the igraph interface for *Mathematica*

This notebook can be opened using the command `IGDocumentation[]`

or through the Documentation Centre. It cannot be saved, so feel free to edit and evaluate input cells, and experiment!

The documentation is currently incomplete. *Contributions are very welcome!*

IGraph/M provides a *Mathematica* interface to the popular igraph network analysis package, as well as many other functions for working with graphs in *Mathematica*. The purpose of IGraph/M is not to replace *Mathematica*’s built-in graph theory functionality, but to complement it. Thus the IGraph/M interface is designed to interoperate seamlessly with built-in functions and datatypes, while also being familiar to users of other igraph interfaces (R, Python or C).

The full igraph functionality is not yet exposed. Priority is given to functionality that is not currently built into *Mathematica*. While many of the functions that IGraph/M provides overlap with built-in ones, like `IGBetweenness`

and `BetweeneessCentrality`

, there are usually some relevant differences. For example, `IGBetweenness`

uses edge weights, while the built-in function `BetweennessCentrality`

does not.

The package can be loaded using

The list of included functions can be queried with the command below. Notice that their names always have the `IG`

prefix. Click on the name of a function to see its usage message.

Or just type a question mark followed by the symbol’s name:

IGraph/M functions work directly with *Mathematica*’s built-in `Graph`

datatype. No new special graph datatype is introduced.

Let’s take a look at a few examples. Let us first generate a graph using the built-in functions of *Mathematica*.

We can compute the betweenness centrality of each vertex either using IGraph/M, …

`{1118.26, 1058.15, 540.601, 127.365, 1175.53, 678.175, 206.929, \ 128.576, 204.019, 535.316, 487.858, 391.669, 0., 135.039, 0., \ 52.5324, 104.28, 12.2286, 75.8798, 110.155, 68.8282, 13.9095, \ 46.4209, 99.3299, 0., 168.196, 213.871, 358.855, 0., 64.9572, \ 5.12619, 102.369, 17.978, 15.569, 95.7266, 8.45843, 25.4984, 13.0274, \ 0., 71.2012, 47.2895, 32.4444, 8.20833, 0., 27.0286, 10.9357, \ 4.60238, 0., 14.7095, 24.7944, 79.125, 7.38301, 22.0817, 43.9635, \ 11.7135, 10.9952, 40.8782, 11.2429, 0., 60.0431, 9.36667, 32.4529, \ 85.4487, 100.431, 15.205, 93.2876, 60.0548, 9.2, 0., 0., 10.512, \ 9.37438, 8.42222, 45.7937, 3.61667, 9.23333, 53.3897, 11.4012, \ 22.0959, 5.24091, 10.2647, 8.66017, 9.97438, 11.0429, 15.8765, \ 12.7798, 0., 30.1744, 0., 0., 4.0373, 9.7, 1., 10.4883, 0., 0., \ 13.7861, 13.8594, 1.7, 2.80952}`

… or using *Mathematica*’s built-ins, and obtain the same result.

`{1118.26, 1058.15, 540.601, 127.365, 1175.53, 678.175, 206.929, \ 128.576, 204.019, 535.316, 487.858, 391.669, 0., 135.039, 0., \ 52.5324, 104.28, 12.2286, 75.8798, 110.155, 68.8282, 13.9095, \ 46.4209, 99.3299, 0., 168.196, 213.871, 358.855, 0., 64.9572, \ 5.12619, 102.369, 17.978, 15.569, 95.7266, 8.45843, 25.4984, 13.0274, \ 0., 71.2012, 47.2895, 32.4444, 8.20833, 0., 27.0286, 10.9357, \ 4.60238, 0., 14.7095, 24.7944, 79.125, 7.38301, 22.0817, 43.9635, \ 11.7135, 10.9952, 40.8782, 11.2429, 0., 60.0431, 9.36667, 32.4529, \ 85.4487, 100.431, 15.205, 93.2876, 60.0548, 9.2, 0., 0., 10.512, \ 9.37438, 8.42222, 45.7937, 3.61667, 9.23333, 53.3897, 11.4012, \ 22.0959, 5.24091, 10.2647, 8.66017, 9.97438, 11.0429, 15.8765, \ 12.7798, 0., 30.1744, 0., 0., 4.0373, 9.7, 1., 10.4883, 0., 0., \ 13.7861, 13.8594, 1.7, 2.80952}`

Let us now assign weights to the edges. Many IGraph/M functions, including `IGBetweenness`

, support edge weights.

`{1569., 1509., 697., 506., 1510., 948., 173., 0., 106., 827., 663., \ 379., 0., 318., 0., 360., 0., 0., 83., 129., 1., 0., 227., 582., 0., \ 91., 236., 213., 0., 60., 0., 334., 1., 53., 549., 0., 0., 0., 0., \ 10., 0., 0., 0., 0., 68., 68., 17., 357., 27., 16., 80., 0., 0., 0., \ 437., 0., 0., 0., 52., 22., 0., 0., 62., 139., 93., 187., 1., 7., 0., \ 0., 0., 16., 0., 69., 10., 98., 0., 1., 4., 21., 0., 0., 0., 0., 0., \ 0., 0., 43., 0., 0., 98., 0., 0., 0., 0., 0., 0., 63., 25., 4.}`

Notice that *Mathematica* 12.0 does not include functionality to compute weighted vertex betweenness. The built-in function `BetweennessCentrality[]`

ignores the weights.

`{1118.26, 1058.15, 540.601, 127.365, 1175.53, 678.175, 206.929, \ 128.576, 204.019, 535.316, 487.858, 391.669, 0., 135.039, 0., \ 52.5324, 104.28, 12.2286, 75.8798, 110.155, 68.8282, 13.9095, \ 46.4209, 99.3299, 0., 168.196, 213.871, 358.855, 0., 64.9572, \ 5.12619, 102.369, 17.978, 15.569, 95.7266, 8.45843, 25.4984, 13.0274, \ 0., 71.2012, 47.2895, 32.4444, 8.20833, 0., 27.0286, 10.9357, \ 4.60238, 0., 14.7095, 24.7944, 79.125, 7.38301, 22.0817, 43.9635, \ 11.7135, 10.9952, 40.8782, 11.2429, 0., 60.0431, 9.36667, 32.4529, \ 85.4487, 100.431, 15.205, 93.2876, 60.0548, 9.2, 0., 0., 10.512, \ 9.37438, 8.42222, 45.7937, 3.61667, 9.23333, 53.3897, 11.4012, \ 22.0959, 5.24091, 10.2647, 8.66017, 9.97438, 11.0429, 15.8765, \ 12.7798, 0., 30.1744, 0., 0., 4.0373, 9.7, 1., 10.4883, 0., 0., \ 13.7861, 13.8594, 1.7, 2.80952}`

Let us delete the minimum feedback edge set to obtain an acyclic graph:

And try out a few of igraph’s layout algorithms.

Layout functions typically have many options to tune:

Increasing the number of iterations will usually improve the result.

**A final note**

Please refer to the usage messages for information on how to use each function. For more information on the meaning of various function options, the algorithms used by the functions, references, etc. please refer to the C/igraph documentation. The igraph documentation provides article references for most nontrivial algorithms.

The following sections provide general information on each functionality area, and show common usage patterns.

All the graph creation functions in IGraph/M take any standard *Mathematica* Graph option such as `VertexLabels`

, `EdgeLabels`

, `VertexStyle`

, `GraphStyle`

, `PlotTheme`

, etc.

Many included graph creation functions implement random graph models. These use igraph’s (not *Mathematica*’s) random number generator, which can be re-seeded using `IGSeedRandom[]`

.

`IGShorthand`

provides an easy way to create small graphs from a simple and quick-to-type notation.

The available options are:

`SelfLoops -> True`

keeps self-loops in the graph.`MultiEdges -> True`

keeps parallel edges in the graph.

Construct a cycle graph.

Vertex labels are shown by default. They can be turned off using `VertexLabels -> None`

.

The interpretation of `-`

as directed or undirected is controlled by the `DirectedEdges`

option.

Directed edges can be input using `->`

, `<-`

or `<->`

.

`<->`

is interpreted as a pair of directed edges.

Mixed graphs, containing both directed and undirected edges, are supported. Note that mixed graphs are not allowed as input to most IGraph/M functions.

Disconnected components are separated by commas.

Groups of vertices can be given using the colon separator. Edges will be connected to each vertex in the group. This makes it easy to specify a complete graph …

… or a complete bipartite graph.

Vertex names are taken as strings, except when they can be interpreted as an integer.

Spaces are allowed in vertex names, and edges can be specified using any number of `-`

characters.

Self-loops and parallel edges are removed by default because these are often created as an undesired by-product of vertex groups. They can be re-enabled using the `SelfLoops`

or `MultiEdges`

options when desired.

The vertex order will follow the order of appearance of vertices in the input string. To control the order, simply list vertices at the beginning of the shorthand specification.

`IGEmptyGraph`

is a convenience function for creating graphs with no edges.

Create a null graph.

Create an empty graph on 15 vertices.

The built-in `EmptyGraphQ`

returns `True`

for these graphs.

creates a graph based on the LCF notation.

The Möbius–Kantor graph is `[5, -5]^8`

.

The Pappus graph is `[5, 7, -7, 7, -7, -5]^3`

.

The cuboctahedral graph is `[4, 2]^6`

.

`IGChordalRing[n, w]`

constructs an extended chordal ring based on the offset specification vector or matrix \(w\) as follows:

It creates a cycle graph (i.e. ring) on \(n\) vertices.

For each vertex \(i\) on the ring, it adds a chord to a vertex \(w[[(i \bmod p)]]\) steps ahead counter-clockwise on the ring.

If \(w\) is a matrix, the procedure is carried out for each row.

The number of vertices \(n\) must be an integer multiple of the number of columns in the matrix \(w\).

The available options are:

`DirectedEdges -> True`

creates a graph with directed edges.`SelfLoops -> False`

prevents the creation of self-loops.`MultiEgdes -> False`

prevents the creation of multi-edges.

Create an extended chordal graph.

Negative offsets are allowed.

`IGChordalGraph`

may create self-loops or multi-edges. This can be prevented by setting the `SelfLoops`

or `MultiEdges`

options to `False`

.

Create a chordal graph with directed edges.

Colour the chords of the ring based on which entry of the \(w\) vector they correspond to.

```
w = {2, 3, 4};
IGChordalRing[12, w, GraphStyle -> "ThickEdge",
EdgeStyle -> Opacity[1/2]] // IGEdgeMap[
ColorData[97],
EdgeStyle -> Function[g,
Table[
If[i <= VertexCount[g], 0, Mod[i, Length[w], 1]], {i,
EdgeCount[g]}]
]
]
```

creates a square lattice graph of the given dimensions. The available options are:

`"Radius"`

controls the size of the neighbourhood within which vertices will be connected.`"Periodic" -> True`

creates a periodic lattice.`"Mutual" -> True`

inserts directed edges in both directions when`DirectedEdges -> True`

is used.

In previous versions, `IGSquareLattice`

was called `IGMakeLattice`

. This name can still be used as a synonym for the sake of backwards compatibility, however, it will be removed in a future version.

To create other types of lattices, see `IGTriangleLattice`

and `IGLatticeMesh`

.

```
Graph3D@IGSquareLattice[{2, 5}, DirectedEdges -> True,
"Periodic" -> True, PlotTheme -> "NeonColor"]
```

`IGTriangularLattice`

can create a triangular grid graph in the shape of a triangle or a rectangle. To generate other types of lattices, see `IGSquareLattice`

and `IGLatticeMesh`

.

The available options are:

`DirectedEdges -> True`

creates a directed graph.`"Periodic" -> True`

creates a periodic lattice.

Generate a triangular lattice on an equilateral triangle with 6 vertices along each of its edges.

Create a directed triangle lattice on a rectangle. Notice the vertex labelling and that the arrows are oriented from smaller index vertices to larger index ones, making this an acyclic graph.

```
IGTriangularLattice[{4, 4}, DirectedEdges -> True,
VertexShapeFunction -> "Name", PerformanceGoal -> "Quality"]
```

Create a triangle lattice and colour its vertices.

```
IGTriangularLattice[{8, 6}, VertexSize -> Large, EdgeStyle -> Gray] //
IGVertexMap[ColorData[98], VertexStyle -> IGMinimumVertexColoring]
```

Take a hexagonal subgraph of a triangle lattice.

```
g = IGTriangularLattice[13];
center = First@GraphCenter[g];
VertexDelete[g,
Complement[VertexList[g], AdjacencyList[g, center, 4], {center}]
]
```

Create a periodic (i.e. toroidal topology) triangle lattice.

The available options are:

`DirectedEdges -> True`

creates a directed tree.

`IGSymmetricTree`

creates a tree where successive layers (i.e. vertices at the same distance from the root) have the specified number of children.

Create a tree where the root has 4 children, its children have 3 children, and so on.

Create a directed tree.

`IGSymmetricTree`

is guaranteed to label vertices in breadth-first order. Deeper layers have higher integer labels.

A Bethe lattice is an infinite tree in which all vertices have the same degree. `IGBetheLattice[n, k]`

computes the first `n`

layers of such a tree. Each non-leaf vertex will have degree `k`

. The default degree is 3.

`IGBetheLattice`

differs from `CompleteKaryTree`

in that the degree of the root will be the same as the degree of other non-lead nodes.

Generate a tree where non-leaf nodes have degree 5, and use directed edges.

Colour vertices based on their distance from the root (i.e. the “layer” they are part of).

```
IGVertexMap[
ColorData[68],
VertexStyle -> (First@IGDistanceMatrix[#, {1}] &),
IGBetheLattice[5, GraphStyle -> "BasicBlack", VertexSize -> 0.4]
]
```

A Prüfer sequence is a unique representation of an \(n\)-vertex labelled tree as \(n-2\) integers between \(1\) and \(n\).

Use `IGToPrufer`

to convert a tree back to its Prüfer sequence.

Generate all labelled trees on 4 nodes:

Of these, only two are non-isomorphic.

`IGExpressionTree`

constructs the tree graph corresponding to an arbitrary *Mathematica* expression. The vertices of the tree will be the positions of the corresponding subexpressions.

`IGExpressionTree`

takes all standard `Graph`

options. The `VertexLabels`

option takes the following special values:

`VertexLabels -> "Head"`

labels branch vertices with the`Head`

of the corresponding subexpression and leaf vertices with the corresponding atomic expression.`VertexLabels -> "Subexpression"`

labels vertices with the corresponding subexpression.`VertexLabels -> "Name"`

labels vertices with their name, i.e. the position of the corresponding subexpression.`VertexLabels -> None`

uses no labels.

`IGExpressionTree`

constructs a graph corresponding to the structure of a *Mathematica* expression.

The expression tree is similar to what `TreeForm`

displays, but unlike `TreeForm`

’s output, it is a `Graph`

object that works with all graph functions.

The vertex names are the position specifications of the corresponding subexpressions.

Place the vertex labels in the centre and construct an undirected graph.

```
IGExpressionTree[x^2 + y^2,
GraphStyle -> "DiagramGold",
VertexLabels -> Placed["Head", Center], VertexSize -> Large
]
```

Create an undirected graph, labelled with subexpressions.

Certain trees are easier to construct through their corresponding nested expression.

An equivalent of `IGSymmetricTree`

can be easily implemented using `IGExpressionTree`

.

```
IGExpressionTree[ConstantArray[1, {3, 4, 5}], VertexLabels -> None,
GraphLayout -> "RadialEmbedding"]
```

Define a tree through a substitution system.

```
IGExpressionTree[
Nest[ReplaceAll[{0 -> {0, 1}, 1 -> {0}}], {0, 1}, 3],
VertexLabels -> None, GraphStyle -> "VibrantColor"
]
```

To format each node so that it fits a label, it is necessary to set an explicit `VertexShapeFunction`

.

```
IGExpressionTree[First@Roots[x^2 + a x + 1 == 0, x],
VertexLabels -> "Subexpression",
PerformanceGoal -> "Quality",
ImageSize -> 280
] //
IGVertexMap[
Function[e, Inset[Panel[e], #1] &],
VertexShapeFunction -> IGVertexProp[VertexLabels]
] // RemoveProperty[#, VertexLabels] &
```

The available options are:

`DirectedEdges -> True`

creates a directed graph.`SelfLoops -> True`

includes self-loops.

Create an undirected complete graph with loops.

Create a directed complete graph with loops.

Create a list of complete graphs starting with the null graph.

Create a complete graph on the given vertices.

Create a complete acyclic directed graph on 5 vertices.

Create a complete acyclic graph on the given vertices. The directed edges always run from vertices that appear earlier in the list to those that appear later.

The vertices of the Kautz graph \(K_m^n\) are strings of length \(n+1\), composed of \(m+1\) distinct symbols, with the restriction that two adjacent symbols in the string may not be the same. A vertex \(s_1s_2\text{$\ldots $s}_ns_{n+1}\) connects to all other vertices of the form \(s_2\text{$\ldots $s}_{n+1}x\), where \(x\) can be any symbol distinct from \(s_{n+1}\).

The Kautz graph \(K_m^n\) has \((m+1) m^n\) vertices, with each vertex having in-degree and out-degree \(m\). Therefore, it has \((m+1) m^{n+1}\) edges.

The line graph of Kautz graph \(K_m^n\) is \(K_m^{n+1}\).

Visualize the Kautz graph \(K_2^3\) on 3 characters with string length 4 in three dimensions.

Label the vertices of the Kautz graph on 3 characters with string length 2.

```
labels = StringJoin /@
DeleteCases[Tuples[{"A", "B", "C"}, {2}], {c_, c_}];
IGKautzGraph[2, 1,
VertexLabels -> Thread[Range[6] -> (Placed[#, Center] &) /@ labels],
VertexSize -> Large, VertexShapeFunction -> "Capsule",
PerformanceGoal -> "Quality",
PlotTheme -> "CoolColor", VertexLabelStyle -> White
]
```

This function uses the Havel–Hakimi algorithm (undirected case) or Kleitman–Wang algorithm (directed case) to construct a graph with the given degree sequence. These algorithms work by selecting a vertex, and connecting up all its free (out-)degrees to other vertices with the largest degrees. In the directed case, the “largest” degrees are determined by lexicographic ordering of (in, out)-degree pairs. The order in which vertices are selected is controlled by the `Method`

option.

To randomly sample multiple realizations of a degree sequence, use `IGDegreeSequenceGame`

.

Available `Method`

option values:

`"SmallestFirst"`

will choose a smallest-degree vertex in each step of the algorithm. This results in a disassortative network. In the undirected case, this method is guaranteed to construct a connected graph, if one exists. See http://szhorvat.net/pelican/hh-connected-graphs.html for the proof. In the directed case, it tends to construct weakly-connected graphs, but this is not guaranteed.`"LargestFirst"`

will choose a largest-degree vertex. This results in an assortative network. This method tends to construct disconnected graphs. This is the most common variant of the Havel–Hakimi algorithm implemented in other packages.`"Index"`

will choose vertices in the order of their indices.

Create a directed graph.

This function is provided for convenience for those who have the book *An Atlas of Graphs* by Ronald C. Read and Robin J. Wilson, and for those who wish to replicate results obtained with other packages that include this database. For all other purposes, use *Mathematica*’s built-in GraphData function.

Retrieve graph number 789:

`IGFromNauty`

converts a Graph6, Digraph6 or Sparse6 string to a graph. These formats originate with the nauty suite of programs and are supported by many other graph theory software.

Interpret a Graph6 string.

Interpret a Sparse6 string. These start with a `:`

character.

Interpret a Digraph6 string. These start with a `&`

character.

`IGFromNauty`

does not support headers or whitespace in the string. To handle these, or to interpret a multiline string, use `IGImportString[…, "Nauty"]`

.

These graph creation functions use igraph’s random graph generator, which can be seeded using `IGSeedRandom`

.

`IGErdosRenyiGameGNM`

uniformly samples graphs with \(n\) vertices and \(m\) edges. This random graph model is known as the Erdős–Rényi \(G(n,m)\) model.

In `IGErdosRenyiGameGNP`

, each edge is present with the same and independent probability. This model is known as the Erdős–Rényi \(G(n,p)\) model or Gilbert model.

The available options are:

`DirectedEdges -> True`

produces a directed graph.`SelfLoops -> True`

allows self-loops.

Create a random graph with 10 vertices and 20 edges.

Create a directed graph and allow self-loops.

Insert each edge with a probability of 20%.

The \(G(n,p)\) model produces connected graphs with high probability for \(p>\ln (n)/n\).

```
n = 300;
ListPlot[
Table[
{p, Mean@
Boole@Table[ConnectedGraphQ@IGErdosRenyiGameGNP[n, p], {50}]},
{p, 0, 0.05, 0.0005}
],
GridLines -> {{Log[n]/n}, None}
]
```

`IGBipartiteGameGNM`

and `IGBipartiteGameGNP`

are equivalent to `IGErdosRenyiGNM`

and `IGErdosRenyiGNP`

, but they generate bipartite graphs.

The available options are:

`DirectedEdges -> True`

creates a directed graph.`"Bidirectional" -> True`

allows directed edges to run in either direction between the two partitions. The default is`False`

, which means that edges will run only from the first partition to the second. This option is ignored for undirected graphs.

Create a bipartite directed graph with edges running either uni-directionally or bidirectionally between the two partitions.

```
IGLayoutBipartite@
IGBipartiteGameGNM[10, 10, 30, DirectedEdges -> True,
"Bidirectional" -> #] & /@ {False, True}
```

`IGTreeGame`

samples uniformly from the set of labelled trees.

Available options:

`DirectedEdges -> True`

will create a directed tree, with edges oriented away from the root.`Method`

can be used to choose the tree generation algorithm. All methods sample labelled trees uniformly.

Available `Method`

options:

`"PruferCode"`

, works by generating a random Prüfer sequence, then constructing a tree from it. It does not currently support directed trees.`"LoopErasedRandomWalk"`

, uses a loop-erased random walk to uniformly sample the spanning trees of the complete graph.

There are several distinct labellings of isomorphic trees. All of these are generated with equal probability.

Generate directed trees.

Generate a random sparse connected graph by first creating a tree, then adding cycle edges. Note that this method does not sample connected graphs uniformly.

```
randomConnected[nodeCount_, edgeCount_] :=
Module[{tree},
tree = IGTreeGame[nodeCount];
EdgeAdd[tree,
RandomSample[EdgeList@GraphComplement[tree],
edgeCount - nodeCount + 1]]
]
```

Colour the nodes of a random tree by their inverse average distance to other nodes.

```
IGVertexMap[
ColorData["SolarColors"],
VertexStyle -> Rescale@*IGCloseness,
IGTreeGame[1000, Background -> Black, ImageSize -> Large,
EdgeStyle -> LightGray]
]
```

`IGDegreeSequenceGame`

implements various random sampling methods for graphs with a given degree sequence. To quickly construct a single realization of a degree sequence, use `IGRealizeDegreeSequence`

.

`IGDegreeSequenceGame`

takes the following values for its `Method`

option:

`"ConfigurationModel"`

implements the configuration model: it connects up vertex stubs randomly. It may generate both self-loops and multi-edges. Undirected graphs are generated with probability proportional to \(\left(\prod _{i<j}A_{ij}!\prod _iA_{ii}\text{!!}\right){}^{-1}\), where \(A\) is the adjacency matrix, having*twice*the number of loops for each vertex on the diagonal. Directed ones are generated with probability proportional to \(\left(\prod _{i,j}A_{ij}!\right){}^{-1}\).All simple graphs are generated with the same probability, but the probability of multigraphs and graphs with self-loops differs from that of simple graphs and depends on their specific structure.

`"ConfigurationModelSimple"`

also implements the configuration model, but it rejects non-simple graphs. It samples uniformly from the set of all simple graphs with the given degree sequence. This method can be very slow for dense graphs.`"FastSimple"`

is a fast generation algorithm that avoids self-loops and multi-edges. This method can generate any simple graph with the given degree sequence, but it does not sample them uniformly.`"VigerLatapy"`

can sample undirected, connected simple graphs uniformly and uses Monte-Carlo methods to randomize the graphs. This generator should be favoured if undirected and connected graphs are to be generated and execution time is not a concern. igraph uses the original implementation of Fabien Viger; see https://www-complexnetworks.lip6.fr/~latapy/FV/generation.html and the corresponding paper at https://arxiv.org/abs/cs/0502085.

The default method is `"FastSimple"`

. Note that it does not sample uniformly.

The configuration model algorithm is too slow to construct even small dense graphs.

Graphs that are almost complete can be sampled by generating the complement first.

In a \(k\)-regular graph all vertices have degree \(k\). The current implementation is able to generate any \(k\)-regular graph, but it does not sample them with precisely the same probability.

The available options are:

`DirectedEdges -> True`

creates a directed graph.`MultiEdges -> True`

allows the creation of parallel edges.

Not all parameters are valid:

There are no graphs with 5 vertices each having degree 3.

`IGGrowingGame[n, k]`

creates a random graph by successively adding vertices to the graph until the vertex count `n`

is reached. At each step, `k`

new edges are added as well.

The available options are:

`DirectedEdges -> True`

creates a directed graph.`"Citation" -> True`

connects newly added edges to the newly added vertex.

With `"Citation" -> True`

, the newly added edges are connected to the newly added vertices.

Note that while this model can be used to generate random trees, it will not sample them uniformly. If uniform sampling is desired, use `IGTreeGame`

instead.

Create a directed citation graph.

`IGBarabasiAlbertGame`

implements a preferential attachment model. It generates a graph by sequentially adding new vertices with the specified number of edges (\(k\)). The edges will connect to existing vertices with probability \(d^{\beta }+A\), where \(d\) is the in-degree of the existing vertex. The default parameters are \(\beta =1\) and \(A=1\).

The available options are:

`DirectedEdges -> False`

creates an undirected graph.`"TotalDegreeAttraction" -> True`

computes the attachment probability based on the the total degree of existing vertices (i.e. the sum of in- and out-degrees), not their in-degree. Always assumed to be`True`

when using`DirectedEdges -> True`

.`"StartingGraph" -> g`

will use graph`g`

as the starting point for building the preferential attachment graph. The vertex names of`g`

are ignored; the result always uses positive integers as vertex names.

Available `Method`

option values:

`"Bag"`

works by putting the IDs of the vertices into a bag exactly as many times as their (in-)degree, plus once more. Then the required number of cited vertices are drawn from the bag, with replacement. This method might generate multi-edges. It only works if \(\beta =1\) and \(A=1\).`"PSumTree"`

uses a partial prefix-sum tree to generate the graph. It does not generate multi-edges and works for any \(\beta\) and \(A\) values.`"PSumTreeMultiple"`

works like`"PSumTree"`

but allows multi-edges.

The built-in `BarabasiAlbertGraphDistribution`

is equivalent to using \(A=0\) and `DirectedEdges -> False`

in `IGBarabasiAlbertGame`

.

Use attachment probability proportional to `degree^1.5 + 1`

.

The `"Bag"`

method may generate parallel edges:

Create a graph with the given out-degree sequence. The \(k^{\text{th}}\) entry in the degree sequence list must be no greater than \(k\).

Create a preferential attachment graph using a 4-node complete graph as the starting point.

The two-argument form produces results equivalent to that of the built-in `WattsStrogatzGraphDistribution`

.

The extended form allows for multi-dimensional lattices. Create a graph by randomly rewiring a two-dimensional toroidal lattice of \(10\times 10\) nodes:

`IGStaticFitnessGame`

generates a random graph by connecting vertices based on their fitness score. The algorithm starts with \(n\) vertices and no edges. Two vertices are selected with probabilities proportional to their fitness scores (for directed graphs, a starting vertex is selected based on its out-fitness and an end vertex based on its in-fitness). If they are not yet connected, an edge is inserted between them. The procedure is repeated until the number of edges reaches \(m\).

The expected degree of each vertex is proportional to its fitness score. This is exactly true when self-loops and multi-edges are allowed, and approximately true otherwise.

`IGStaticFitnessGame`

approximates the Chung–Lu model in which each edge `i \[UndirectedEdge] j`

is present independently, with probability

\[p_{ij}= \begin{array}{ll} \{ & \begin{array}{ll} \frac{f_if_j}{2m} & \text{if} i\neq j \\ \frac{f_if_j}{4m} & \text{if} i=j \\ \end{array} \\ \end{array} ,\]

where \(m=\frac{1}{2}\sum _kf_k\).

Unlike the Chung–Lu algorithm, which would require \(O\left(m^2\right)\) computation steps, `IGStaticFitnessGame`

runs in \(O(m)\) time.

The available options are:

`SelfLoops -> True`

allows the creation of self-loops.`MultiEdges -> True`

allows the creation of parallel edges.

Create an undirected graph with four high-degree nodes and 40 low-degree ones.

```
weights = Join[{10, 10, 10, 10}, ConstantArray[1, 40]];
IGStaticFitnessGame[Total[weights]/2, weights]
```

Create a directed graph.

When self-loops and multi-edges are allowed, the expected degree of each vertex is proportional to its fitness score.

```
degrees = {3, 3, 2, 2, 2, 1, 1};
Table[
VertexDegree@IGStaticFitnessGame[
Total[degrees]/2, degrees,
SelfLoops -> True, MultiEdges -> True
],
{1000}
] // N // Mean
```

When generating simple graphs, this holds only approximately.

```
degrees = {3, 3, 2, 2, 2, 1, 1};
Table[
VertexDegree@IGStaticFitnessGame[
Total[degrees]/2, degrees
],
{1000}
] // N // Mean
```

`IGStaticPowerLawGame`

generates a directed or undirected random graph where the degrees of vertices follow power-law distributions with prescribed exponents. For directed graphs, the exponents of the in- and out-degree distributions may be specified separately.

This function is equivalent to `IGStaticFitnessGame`

with a fitness vector \(f\) where \(f_i=\frac{1}{\text{exponent}-1}\).

Note that significant finite size effects may be observed for exponents smaller than 3 in the original formulation of the game. This function removes the finite size effects by default by assuming that the fitness of vertex \(i\) is \(\left(i+i_0-1\right){}^{-\alpha }\), where \(i_0\) is a constant chosen appropriately to ensure that the maximum degree is less than the square root of the number of edges times the average degree; see the paper of Chung and Lu, and Cho et al. for more details.

The available options are:

`SelfLoops -> True`

allows the creation of self-loops.`MultiEdges -> True`

allows the creation of parallel edges.`"FiniteSizeCorrection" -> False`

disables finite size correction, which is used by default.

Create a graph with a power-law degree distribution of exponent 2.5.

Create a directed graph with power-law in- and out-degree distributions.

Goh K-I, Kahng B, Kim D: Universal behaviour of load distribution in scale-free networks. Phys Rev Lett 87(27):278701, 2001.

Chung F and Lu L: Connected components in a random graph with given degree sequences. Annals of Combinatorics 6, 125-145, 2002.

Cho YS, Kim JS, Park J, Kahng B, Kim D: Percolation transitions in scale-free networks under the Achlioptas process. Phys. Rev. Lett. 103:135702, 2009.

The `ratesMatrix`

argument gives the connection probability between and within blocks (groups of vertices). The `blockSizes`

argument gives the size of each block (vertex group).

The available options are:

`DirectedEdges -> True`

creates a directed graph.`SelfLoops -> True`

allows the creation of self-loops.

The forest fire model is a growing graph model. In every time step, a new vertex is added to the graph. The new vertex chooses the specified number of ambassadors (default: 1) and starts a simulated forest fire at their locations. The fire spreads through the directed edges. The spreading probability along an edge is given by `pForward`

. The fire may also spread backwards on an edge with probability `pForward * rBackward`

. When the fire has ended, the newly added vertex connects to all the vertices that were burned in the fire.

The forest fire model intends to reproduce the following network characteristics, observed in real networks:

Heavy-tailed in-degree and out-degree distributions.

Community structure.

Densification power-law. The network is densifying in time, according to a power-law rule.

Shrinking diameter. The diameter of the network decreases in time.

The available options are:

`DirectedEdges -> False`

generates an undirected graph.

Generate a graph with only forward burning.

Generate a graph from the forest fire model, and visualize its community structure.

```
IGForestFireGame[100, 0.2, 1, 2, DirectedEdges -> False,
GraphLayout -> {"EdgeLayout" -> "HierarchicalEdgeBundling"}]
```

Plot the cumulative in-degree distribution for different backward to forward burning probability ratios.

```
Table[
Histogram[
VertexInDegree@
IGForestFireGame[2000, 0.4, r, 2, DirectedEdges -> True],
"Log", {"Log", "SurvivalCount"},
PlotLabel -> Row[{"r=", r}]
],
{r, 0, 0.8, 0.2}
]
```

Jure Leskovec, Jon Kleinberg and Christos Faloutsos. Graph evolution: Densification and shrinking diameters.

*ACM Transactions on Knowledge Discovery from Data (TKDD)*, 2007. https://doi.org/10.1145/1217299.1217301Jure Leskovec, Jon Kleinberg and Christos Faloutsos. Graphs over time: densification laws, shrinking diameters and possible explanations.

*KDD ’05: Proceeding of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining*, 177–187, 2005.

This function simulates a growing random graph according to the following algorithm:

At each time step, a new vertex is added. Its type is randomly selected according to the type weights. Then `k`

existing pairs of vertices are selected randomly, and each pair attempts to connect. The probability of success for given types of vertices is given by the preference matrix.

This algorithm may create self-loops and multi-edges.

The available options are:

`DirectedEdges -> True`

creates a directed graph.

This function simulates a growing random graph according to the following algorithm:

At each time step, a new vertex is added. Its type is randomly selected according to the type weights. It attempts to connect to `k`

distinct existing vertices. The probability of success for given types of vertices is given by the preference matrix.

The available options are:

`DirectedEdges -> True`

creates a directed graph.

Available options:

`"Periodic" -> True`

assumes a toroidal topology

Use a toroidal topology and draw “wraparound” edges with dashed lines.

```
IGGeometricGame[50, 0.2, "Periodic" -> True] //
IGEdgeMap[
If[EuclideanDistance @@ # > 0.2, Dashed, None] &,
EdgeStyle -> IGEdgeVertexProp[VertexCoordinates]
]
```

`IGRewire`

will try to rewire the edges of the graph the given number of times by switching random pairs of edges as below, thus preserving the graph’s degree sequence.

⟶ or ⟶

The switches succeed only if they would not create multi-edges. The parameter \(n\) specifies the number of switch attempts, not the number of successful switches.

For directed graphs, the switches are such that they preserve both the in- and out-degree sequence.

The vertex ordering of the graph is retained.

**Warning:** Most graph properties, such as edge weights, will be lost.

The available options are:

`SelfLoops -> True`

allows the creation of self-loops.

Generate a random network with scale-free degree distribution:

Use `SelfLoops -> True`

to allow creating loops.

`IGRewrire`

never creates any multi-edges. Multigraphs are allowed as input, but a warning is given.

Uniformly sample simple labelled graphs with a given degree sequence by first creating a single realization, then rewiring it a sufficient amount of times.

```
Table[
IGRewire[IGRealizeDegreeSequence[degseq], 100],
{1000}
] // CountsBy[AdjacencyMatrix] // KeySort //
KeyMap[AdjacencyGraph[#, VertexShapeFunction -> "Name"] &]
```

`IGRewireEdges`

randomly rewires each edge of the graph with the given probability. The vertex ordering is retained.

For directed graphs, it can optionally rewire only the starting point or endpoint of directed edges, thus preserving the out- or in-degree sequence. In this case, the `MultiEdges`

option is ignored and multi-edges may be created.

**Warning:** Most graph properties, such as edge weights, will be lost.

The available options are:

`SelfLoops -> True`

allows the creation of self-loops.`MultiEdges -> True`

allows the creation of multi-edges.

Create a random graph with 10 vertices and 20 edges, while allowing for multi-edges:

Rewire the endpoint of each edge, preserving the out-degree sequence.

Note that multi-edges were created.

`IGVertexContract[g, {set1, set2, …}]`

will simultaneously contract multiple vertex sets into single vertices.

The name of a contracted vertex will be the same as the first element of the corresponding set. Vertex ordering is not retained. Edge ordering is retained only when using *both* `SelfLoops -> True`

and `MultiEdges -> True`

.

**Warning:** Most graph properties, such as edge weights, will be lost.

The available options are:

`SelfLoops -> True`

keeps any self-loops created during contraction.`MultiEdges -> True`

keeps any parallel edges created during contraction.

When using both `SelfLoops -> True`

and `MultiEdges -> True`

, the edge ordering is maintained relative to the input graph. This allows easily transferring edge weights, and combining them if necessary.

```
IGWeightedSimpleGraph[
IGVertexContract[g, {{"a", "b"}},
SelfLoops -> True, MultiEdges -> True,
EdgeWeight -> IGEdgeProp[EdgeWeight][g]
],
EdgeLabels -> "EdgeWeight", VertexLabels -> "Name"
]
```

`IGConnectNeighborhood[g, k]`

connects each vertex in `g`

to its order `k`

neighbourhood. This operation is also known as the \(k^{\text{th}}\) power of the graph.

`IGConnectNeighborhood`

differs from the built-in `GraphPower`

in that it preserves parallel edges and self-loops.

**Warning:** Most graph properties, such as edge weights, will be lost.

Connect each vertex to its second order neighbourhood:

Connect each vertex to its third order neighbourhood:

`IGMycielskian`

applies the Mycielski construction to an undirected graph on \(n\geq 2\) vertices to obtain a larger graph (the *Mycielskian*) on \(2 n+1\) vertices. If the graph has less than 2 vertices, then instead of applying the standard Mycielski construction, `IGMycielskian`

simply adds one vertex and one edge.

If the original graph has chromatic number \(k\), its Mycielskian has chromatic number \(k+1\). The Mycielski construction preserves the triangle-free property of the graph.

Construct triangle-free graphs with successively larger chromatic numbers.

`IGSmoothen`

suppresses all degree-2 vertices, thus obtaining the smallest topologically equivalent (i.e. homeomorphic) graph. See also `IGHomeomorphicQ`

.

The vertex names are preserved, and the weights of merged edges are summed up. All other graph properties are discarded. Edge directions are also discarded.

The smallest topological equivalent of a path graph consists of two connected vertices.

The result may contain self-loops. The smallest topological equivalent of a cycle graph is a single vertex with a self-loop.

The result may also contain multi-edges.

The result is always a weighted graph. When contracting edges, their weights are added up. If the input graph was not weighted, all of its edge weights are considered to be 1. Thus, the graph distance of any two vertices in the result is always the same as it was in the input graph.

The result does not contain any degree-2 vertices.

The vertex coordinates, as well as any other graph properties are discarded.

Vertex coordinates can be transferred to the new graph as follows:

Create a tree in which every non-leaf node has a degree of at least 3.

Let us compute the effective resistance of a resistor network by repeated smoothing (merger of resistors in series) and simplification (merger of resistors in parallel). Resistances are stored as edge weights. A zero-resistance input and output terminal is added to prevent the premature smoothing of these points.

Merge resistors in series …

… then merge resistors in parallel and check the resulting edge weights.

Repeat until a single resistor remains.

Weighted graphs are supported by all betweenness functions in IGraph/M.

The betweenness of a vertex or edge is, roughly speaking, the number of shortest paths passing through it. More formally, the betweenness of vertex \(i\) is \(b_i=\sum _{i\neq s\neq t} \frac{g_{st}^{(i)}}{g_{st}}\), where \(g_{st}\) is the total number of shortest paths (geodesics) between vertices \(s\) and \(t\), and \(g_{st}^{(i)}\) is the number of shortest paths between vertices \(s\) and \(t\) that pass through \(i\).

Available options:

`Method`

, see below. The default is`"Precise"`

. This method is currently only available for vertex betweenness calculations.`Normalized -> True`

will compute the normalized betweenness by dividing the result by the number of (ordered or unordered) vertex pairs used in the shortest path calculation. Thus the normalization factor is \((V-1) (V-2)\) for directed graphs and \(\frac{1}{2} (V-1) (V-2)\) for undirected graphs. The normalized value lies between 0 and 1.

Available `Method`

options for vertex betweenness calculations:

`"Precise"`

is accurate, but slow. This is the default.`"Fast"`

is fast, but may give incorrect results for large graphs with lots of shortest paths.

Compare the `"Fast"`

and `"Precise"`

methods:

For this large grid graph, the `"Fast"`

method no longer gives accurate results:

Visualize the vertex and edge betweenness of a weighted geometrical graph, where weights represent Euclidean distances.

```
pts = RandomPoint[Disk[], 100];
IGMeshGraph[
DelaunayMesh[pts],
EdgeStyle -> Thick, VertexStyle -> EdgeForm[None]
] //
IGVertexMap[
ColorData["SolarColors"],
VertexStyle -> Rescale@*IGBetweenness
] /*
IGEdgeMap[
ColorData["SolarColors"],
EdgeStyle -> Rescale@*IGEdgeBetweenness
]
```

Compute the betweenness of a subset of vertices.

Visualize the betweenness of a periodic grid with slightly randomized edge weights.

```
n = 40;
IGSquareLattice[{n, n},
"Periodic" -> True,
VertexCoordinates -> Tuples[Range[n], {2}],
EdgeWeight -> {_ :> RandomReal[{.99, 1.01}]},
GraphStyle -> "BasicBlack",
EdgeShapeFunction -> None,
VertexSize -> 1
] // IGVertexMap[
ColorData["BlueGreenYellow"],
VertexStyle -> Rescale@*IGBetweenness
]
```

**Possible issues:**

Weighted betweenness calculations may be affected by numerical precision when non-integer weights are used. Betweenness computations count shortest paths, which means that the total weight of different paths must be compared for equality. Equality testing with floating point numbers is unreliable. This can be demonstrated on an example which is purposefully constructed to be problematic.

An unweighted grid graph has many shortest paths between the same pair of vertices.

Let us construct non-integer weights which can add up to (equal) integers in many different ways.

Let us now associate weights with edges and order the edges in two different random ways. Betweenness should not depend on edge ordering, so graphs constructed from both should have the same betweenness values.

```
asc1 = Association@RandomSample@Thread[EdgeList[g] -> weights];
asc2 = Association@RandomSample@Thread[EdgeList[g] -> weights];
```

```
g1 = Graph[Keys[asc1], EdgeWeight -> Values[asc1]];
g2 = Graph[Keys[asc2], EdgeWeight -> Values[asc2]];
```

```
KeySort@AssociationThread[VertexList[g1], IGBetweenness[g1]] -
KeySort@AssociationThread[VertexList[g2], IGBetweenness[g2]]
```

`<|1 -> 0., 2 -> 0., 3 -> 0., 4 -> 0., 5 -> 0., 6 -> 0., 7 -> 0., 8 -> 0., 9 -> 0., 10 -> -7.10543*10^-15, 11 -> 0., 12 -> 0., 13 -> 0., 14 -> 0., 15 -> -2.84217*10^-14, 16 -> -1.42109*10^-14, 17 -> 0., 18 -> 0., 19 -> 0., 20 -> 0., 21 -> -2.84217*10^-14, 22 -> 0., 23 -> 0., 24 -> 0., 25 -> 0., 26 -> 0., 27 -> 0., 28 -> 0., 29 -> 0., 30 -> 0., 31 -> 0., 32 -> 0., 33 -> 0., 34 -> 0., 35 -> 0., 36 -> 0.|>`

Yet the results differ. This is because IGraph/M works with floating point numbers. Even when different sums of *exact* weights happen to be equal, floating point calculations will give slightly different results.

To obtain a reliable result, we must use integer weights. The weights in these example were inverses of `{1, 2, 3, 4, 5}`

. Multiplying these by their least common multiple will always yield an integer.

```
asc1 = Association@RandomSample@Thread[EdgeList[g] -> 60 weights];
asc2 = Association@RandomSample@Thread[EdgeList[g] -> 60 weights];
```

```
g1 = Graph[Keys[asc1], EdgeWeight -> Values[asc1]];
g2 = Graph[Keys[asc2], EdgeWeight -> Values[asc2]];
```

```
KeySort@AssociationThread[VertexList[g1], IGBetweenness[g1]] -
KeySort@AssociationThread[VertexList[g2], IGBetweenness[g2]]
```

`<|1 -> 0., 2 -> 0., 3 -> 0., 4 -> 7.10543*10^-15, 5 -> 0., 6 -> 0., 7 -> 0., 8 -> 0., 9 -> 7.10543*10^-15, 10 -> 7.10543*10^-15, 11 -> 0., 12 -> 0., 13 -> 0., 14 -> 0., 15 -> 2.84217*10^-14, 16 -> 4.26326*10^-14, 17 -> 0., 18 -> 0., 19 -> 0., 20 -> 0., 21 -> 2.84217*10^-14, 22 -> 7.10543*10^-15, 23 -> 0., 24 -> 0., 25 -> 0., 26 -> 0., 27 -> 0., 28 -> 0., 29 -> 0., 30 -> 0., 31 -> 0., 32 -> 0., 33 -> 0., 34 -> 0., 35 -> 0., 36 -> 0.|>`

Now `IGBetweenness`

gives the same result regardless of edge ordering.

The normalized closeness centrality of a vertex is the inverse average shortest path length to other vertices.

Weighted graphs are supported.

Available options:

`Normalized -> True`

will normalize values by the number of vertices minus one. Effectively, this computes closeness as the reciprocal of the average distance to all other nodes.`Normalized -> False`

uses the sum (instead of the average) of distances to all other nodes. The default is`False`

for consistency with igraph interfaces in other languages.

igraph’s closeness calculation differs from *Mathematica*’s in that when there is no path connecting two vertices, the total number of vertices is used as their distance (a number larger than any other distance in an unweighted graph). *Mathematica* computes closeness separately within each connected component.

Visualize the closeness of nodes in a weighted geometrical graph where weights correspond to Euclidean distances.

```
pts = RandomPoint[Polygon@CirclePoints[3], 75];
IGVertexMap[
ColorData["Rainbow"],
VertexStyle -> Rescale@*IGCloseness,
IGMeshGraph[DelaunayMesh[pts], GraphStyle -> "BasicBlack"]
]
```

`Indeterminate`

is returned for a single-vertex graph.

Weighted graphs and multigraphs are supported.

The default damping factor is 0.85.

The following `Method`

options are available:

`"PowerIteration"`

, deprecated, not recommended. Takes suboptions`"MaxIterations"`

and`"Epsilon"`

. Does not support weights or multigraphs.`"Arnoldi"`

uses ARPACK`"PRPACK"`

uses PRPACK. It is the default method.

Weighted graphs are supported.

The available options are:

`Normaized -> True`

will scale the result so that the maximum centrality is 1. The default is`True`

.`DirectedEdges -> False`

ignores edge directions.

Weighted graphs are supported.

The available options are:

`Normalized -> True`

scales the result so that the maximum centrality is 1. The default is`True`

.

Weighted graphs are supported.

*Centralization* is computed from centrality values in a way equivalent to `Total[Max[centralities] - centralities]`

. With the (default) option `Normalized -> True`

, the result is normalized by dividing by the highest possible centralization score of any graph of the same directedness on the same number of vertices.

For most centrality types, the highest centralization is achieved by the `StarGraph`

.

In the case of the degree centralization, the highest possible centralization score depends on whether self-loops are allowed. This is controlled by the `SelfLoops`

option of `IGDegreeCentralization`

. The default is `SelfLoops -> True`

.

`IGDirectedAcyclicGraphQ`

tests if a graph is directed and has no directed cycles.

`IGDirectedAcyclicGraphQ`

returns `True`

for graphs with no edges.

`IGTopologicalOrdering`

is to the built-in `TopologicalSort`

as `Ordering`

is to `Sort`

: it returns the permutation which sorts vertices in topological order. When vertices are ordered topologically, all directed edges point from earlier vertices to later ones.

Graphs must be acyclic for topological sorting to be possible.

If the vertices are laid out from left to right in topological order, all edges will point from left to right.

```
Graph[g,
EdgeShapeFunction ->
GraphElementData[{"CurvedEdge", "Curvature" -> 1.5}]
] // IGVertexMap[{#, 0} &,
VertexCoordinates -> IGTopologicalOrdering /* Ordering]
```

When the graph contains cycles, and a complete topological sort cannot be performed, only a partial result is returned.

`IGFeedbackArcSet[]`

returns a set of directed edges (also called *arcs*) the removal of which makes the graph acyclic.

With `Method -> "IntegerProgramming"`

, it finds an exact minimal feedback arc set through integer programming using the triangle inequality formulation. With `Method -> "EadesLinSmyth"`

, it finds a feedback arc set (not necessarily minimal) using the fast “GR” heuristic of Eades, Lin and Smyth (1993).

The following directed graph is not acyclic.

Find a set of edges whose removal breaks all cycles.

Vertices of a directed acyclic graph can be sorted topologically. `IGTopologicalOrdering`

returns a permutation that sorts them this way, and thus makes the graph’s adjacency matrix upper triangular.

- P. Eades, X. Lin, and W. F. Smyth, A fast and effective heuristic for the feedback arc set problem,
*Inf. Process. Lett.***47**, 319 (1993).

A graph is chordal if each of its cycles of four or more nodes has a chord, i.e. an edge joining two non-adjacent vertices in the cycle. Equivalently, all chordless cycles in a chordal graph have at most 3 vertices.

Chordal graphs are also called *rigid circuit graphs* or *triangulated graphs*.

Grid graphs are not chordal because they have chordless 4 cycles.

Adding chords to the 4 cycles makes them chordal.

`IGChordalCompletion`

computes a set of edges that, when added to a graph, make it chordal. The edge set returned is not usually minimal, i.e. some of the edges may not be necessary to create a chordal graph.

The maximum cardinality search algorithm visits the vertices of the graph in such an order so that every time the vertex with the most already visited neighbours is visited next. The visiting order is animated below:

```
Table[
HighlightGraph[
Graph[g, VertexLabels -> "Name"],
Take[seq, -i]
],
{i, 1, 10}
] // ListAnimate
```

Clustering coefficients are measures of the degree to which vertices in a graph tend to cluster together. They are also referred to as *transitivity*, as they measure how often two vertices that are connected through a third one are also directly connected.

All clustering coefficient calculations in IGraph/M ignore edge directions.

The clustering coefficient of an undirected graph is defined as

\[C=\frac{\text{number of closed ordered triplets}}{\text{number of connected ordered triplets}}\]

The available options are:

`"ExcludeIsolates" -> True`

will cause`Indeterminate`

to be returned if the graph has no connected triplets. With the default`"ExcludeIsolates" -> False`

,`0`

is returned.

The following graph has 10 connected ordered triplets, namely {3, 1, 2}, {2, 1, 3}, {1, 2, 3}, {3, 2, 1}, {2, 3, 1}, {2, 3, 4}, {1, 3, 4}, {1, 3, 2}, {4, 3, 2}, {4, 3, 1}. Out of these, only 6 are closed: {1, 3, 2}, {1, 2, 3}, {2, 1, 3}, {2, 3, 1}, {3, 2, 1}, {3, 1, 2}. Thus the clustering coefficient is `6/10 = 0.6`

.

The local clustering coefficient of a vertex is defined as

\[C=\frac{\text{number of connected pairs of neighbours}}{\text{total number of pairs of neighbours}}\]

The available options are:

`"ExcludeIsolates" -> True`

will cause`Indeterminate`

to be returned for degree 0 and degree 1 vertices. With the default`"ExcludeIsolates" -> False`

,`0`

is returned.

In the following graph, vertex 4 has only one neighbour. Thus its local clustering coefficient will be computed as either `0`

or indeterminate depending on the setting for `"ExcludeIsolates"`

.

The available options are:

`"ExcludeIsolates" -> True`

will cause degree 0 and degree 1 vertices to be excluded from the calculation.

With `"ExcludeIsolates" -> True`

, the local clustering coefficient of vertex 4 will be excluded from the calculation of the average.

`IGWeightedClusteringCoefficient`

computes the *weighted* local clustering coefficient, as defined by A. Barrat et al. (2004) http://arxiv.org/abs/cond-mat/0311416 . This function expects a weighted graph as input.

The available options are:

`"ExcludeIsolates" -> True`

will cause`Indeterminate`

to be returned for degree 0 and degree 1 vertices. With the default`"ExcludeIsolates" -> False`

,`0`

is returned.

The length of a path between two vertices is the number of edges the path consists of. Functions that use edge weights consider the path length to be the sum of edge weights along the path.

`IGDistanceMatrix`

takes the following `Method`

options:

`Automatic`

selects a method automatically. As of version 0.3.0,`"Unweighted"`

is selected for unweighted graphs,`"Dijkstra"`

for weighted graphs with only positive weights, and`"Johnson"`

otherwise.`"Unweighted"`

ignores weights`"Dijkstra"`

uses Dijkstra’s algorithm. All weights must be non-negative.`"BellmanFord"`

uses the Bellman–Ford algorithm. Negative weights are supported but all cycles must have a non-negative total weight.`"Johnson"`

uses the Johnson algorithm. Negative weights are supported but all cycles must have a non-negative total weight.

The igraph C core may override explicit method settings when appropriate. For example, if the graph is not weighted, it always uses `"Unweighted"`

.

`IGDistanceCounts[graph]`

counts all-pair *unweighted* shortest path lengths in the graph. counts unweighted shortest path lengths for paths starting at the given vertices.

For weighted path lengths, or to restrict the computation to both certain start and end vertex sets, use `IGDistanceHistogram[]`

.

Compute how the shortest path length distribution changes as we rewire a grid graph `k`

times.

```
Table[
ListPlot[
Normalize[IGDistanceCounts@IGRewire[GridGraph[{50, 50}], k],
Total],
Joined -> True, Filling -> Bottom,
PlotLabel -> StringTemplate["rewiring steps: ``"][k]
],
{k, {0, 5, 10, 20, 50, 100}}
]
```

`IGDistanceHistogram[]`

computes the weighted shortest path length histogram between the specified start and end vertex sets. The start and end vertex sets do not need to be the same. Note that if the graph is undirected, path lengths between `s`

and `t`

will be double counted (from `s -> t`

and `t -> s`

) if `s`

and `t`

appear both in the starting and ending vertex sets.

`IGDistanceHistogram[]`

is useful when the result of `IGDistanceMatrix[]`

(or `GraphDistanceMatrix[]`

) does not fit in memory.

If the graph is unconnected, vertex pairs between which there is no path are excluded from the calculation. This is different from the behaviour of `MeanGraphDistance[]`

, which returns `∞`

in this case.

The diameter of a graph is the length of the longest shortest path between any two vertices.

The available options are:

`Method`

can take the values`"Unweighted"`

,`"Dijkstra"`

or`Automatic`

.`"Dijkstra"`

takes edge weights into account.`Automatic`

chooses based on whether the graph is weighted.`"ByComponents"`

controls how unconnected graphs are handled. If`False`

,`Infinity`

is returned. If`True`

, the largest diameter within a component is returned.

```
HighlightGraph[g, PathGraph@IGFindDiameter[g],
GraphHighlightStyle -> "DehighlightFade", PlotTheme -> "RoyalColor"
]
```

The eccentricity of a vertex is the longest shortest path to any other vertex. IGEccentricity computes the *unweighted* eccentricity of each vertex within the connected component where it is contained.

The radius of a graph is the smallest eccentricity of any of its vertices.

For unconnected graphs, 0 is returned.

`IGVoronoiCells[graph, centers]`

partitions a graph’s vertices into groups based on which given centre vertex they are the closest to. Edge weights are considered for the distance calculations.

Available options:

`"Tiebreaker"`

sets the function used to decide which cell a vertex should belong to if its distance to several different centres is equal. The default is to use the first qualifying cell. Possible useful settings are`First`

,`Last`

,`RandomChoice`

.

In the event of a tie, a vertex is added to the first qualifying cell. The tiebreaker function can be changed as below.

Find Voronoi cells on a grid.

```
g = GridGraph[{10, 10}, VertexSize -> Medium,
GraphStyle -> "BasicBlack"];
centers = RandomSample[VertexList[g], 3];
HighlightGraph[g,
Append[
Subgraph[g, #] & /@ Values@IGVoronoiCells[g, centers],
Style[centers, Black]
],
GraphHighlightStyle -> "DehighlightHide"
]
```

Edge weights are interpreted as distances.

```
g = IGMeshGraph@DelaunayMesh@RandomPoint[Disk[], 200];
centers = RandomSample[VertexList[g], 3];
HighlightGraph[g,
Append[
Subgraph[g, #] & /@ Values@IGVoronoiCells[g, centers],
Style[centers, Black]
],
GraphHighlightStyle -> "DehighlightGray"
]
```

Generate a graph and verify that it is bipartite.

Verify that no edges run between two disjoint vertex subsets of the graph.

Find a bipartite partitioning of a graph.

Ensure that the partitions are returned in such an order that the first one contains vertex 5.

`$Failed`

is returned for non-bipartite graphs.

We can use `IGPartitionsToMembership`

or `IGKVertexColoring[…, 2]`

to obtain a partition index for each vertex.

The following bipartite graph described the relationship between diseases and genes.

Construct a disease-disease and gene-gene network from it.

Compute an incidence matrix. The default partitioning used by `IGBipartiteIncidenceMatrix`

is the one returned by `IGBipartitePartitions`

.

Reconstruct a graph from an incidence matrix.

Compute an incidence matrix using a given partitioning / vertex ordering. It is allowed to pass only a subset of vertices.

Reconstruct the bipartite graph while specifying vertex names.

The bibliographic coupling of two vertices in a directed graph is the number of other vertices they both connect to. The bibliographic coupling matrix can also be obtained using `IGZeroDiagonal[am.am\[Transpose]]`

, where `am`

is the adjacency matrix of the graph.

The co-citation coupling of two vertices in a directed graph is the number of other vertices that connect to both of them. The co-citation coupling matrix can also be obtained using `IGZeroDiagonal[am\[Transpose].am]`

, where `am`

is the adjacency matrix of the graph.

The Dice similarity coefficient of two vertices is twice the number of common neighbours divided by the sum of the degrees of the vertices.

The Jaccard similarity coefficient of two vertices is the number of common neighbours divided by the number of vertices that are neighbours of at least one of the two vertices being considered.

The inverse log-weighted similarity of two vertices is the number of their common neighbours, weighted by the inverse logarithm of their degrees. It is based on the assumption that two vertices should be considered more similar if they share a low-degree common neighbour, since high-degree common neighbours are more likely to appear even by pure chance.

Isolated vertices will have zero similarity to any other vertex. Self-similarities are not calculated.

- Lada A. Adamic and Eytan Adar: Friends and neighbors on the Web,
*Social Networks*, 25(3):211-230, 2003.

- IGConnectedQ and IGWeaklyConnectedQ
- IGConnectedComponentSizes and IGWeaklyConnectedComponentSizes
- Vertex separators
- IGEdgeConnectivity
- IGVertexConnectivity
- IGBiconnectedQ
- IGBiconnectedComponents and IGBiconnectedEdgeComponents
- IGArticulationPoints
- IGBridges
- IGSourceVertexList and IGSinkVertexList
- IGGiantComponent

`IGConnectedQ`

checks if the graph is (strongly) connected. It is equivalent to `ConnectedGraphQ`

. `IGWeaklyConnectedQ`

check if a directed graph is weakly connected. It is equivalent to `WeaklyConnectedGraphQ`

. Both of these functions use the implementation from the core igraph library, and will always be consistent with it for edge cases such as the null graph.

This graph is connected.

This directed graph is only weakly connected.

The null graph is considered connected by convention.

`IGWeaklyConnectedComponentsSizes`

and `IGConnectedComponentSizes`

return the sizes of the graph’s weakly or strongly connected components in decreasing order.

In large graphs, these functions will be faster than the equivalent `Length /@ ConnectedComponents[g]`

.

The emergence of a giant component as the number of edges in a random graph increases.

The number of weakly and strongly connected components versus the number of edges in a random directed graph.

```
Table[
With[{g = RandomGraph[{1000, m}, DirectedEdges -> True]},
{{m, Length@IGWeaklyConnectedComponentSizes[g]}, {m,
Length@IGConnectedComponentSizes[g]}}
],
{m, 5, 3000, 5}
] // Transpose // ListPlot
```

A vertex separator is a set of vertices whose removal disconnects the graph.

Removing any of these vertex sets will disconnect the graph:

Removing Anna, Nora and Larry also disconnects the graph, thus this vertex set is a separator:

But it is not minimal:

`IGMinimumSeparators`

returns only those vertex separators which are of the smallest possible size in the graph. `IGMinimalSeparators`

returns all separators which cannot be made smaller by removing a vertex from them. The former is a subset of the latter.

`IGEdgeConnectivity`

ignores edge weights. To take edge weights into account, use `IGMinimumCutValue`

instead.

Compute the edge connectivity of the dodecahedral graph.

The edge connectivity of the singleton graph is returned as 0.

According to Steinitz’s theorem, the skeleton of every convex polyhedron is a 3-vertex-connected planar graph.

To find the specific vertex sets that disconnect the graph, use `IGMinimumSeparators`

or `IGMinimalSeparators`

.

The vertex connectivity of the singleton graph is returned as 0.

`IGBiconnectedQ`

checks if a graph is biconnected. Edge directions are ignored.

Since `IGBiconnectedComponents`

does not return any isolated vertices, `Length@IGBiconnectedComponents[g] == 1`

cannot be used to check if a graph is biconnected. Use `IGBiconnectedQ`

instead.

The singleton graph is not considered to be biconnected, but the two-vertex complete graph is.

`IGBiconnectedCompoments`

returns the vertices of the maximal biconnected components of the graph. `IGBiconnectedEdgeComponents`

returns the edges of the components. Edge directions are ignored and isolated vertices are excluded.

`IGBiconnectedComponents`

is equivalent to `KVertexConnectedComponents[…, 2]`

, except that isolated vertices are not returned as individual components.

The articulation vertices will be part of more than a single component, thus the biconnected components are not disjoint subsets of the vertex set.

However, each edge is part of precisely one biconnected components.

Thus, visualizing biconnected components is best done by colouring the edges, not the vertices.

`IGArticulationPoints`

finds vertices whose removal increases the number of (weakly) connected components in the graph. Edge directions are ignored.

```
g = Graph[{1 -> 2, 2 -> 3, 3 -> 1, 3 -> 4, 4 -> 5, 5 -> 6, 6 -> 4},
DirectedEdges -> False, VertexLabels -> Automatic]
```

Articulation points are also size-1 separators.

Highlight the articulation points of a cactus graph.

Compute the block-cut tree of a connected graph. The *blocks* are the biconnected components. Together with the articulation vertices they form a bipartite graph, specifically a tree.

```
RelationGraph[
MemberQ,
Join[IGBiconnectedComponents[g], IGArticulationPoints[g]],
DirectedEdges -> False,
GraphStyle -> "ClassicDiagram",
VertexSize -> {3, 1}/7, VertexLabelStyle -> 8
]
```

A bridge is an edge whose removal disconnects the graph (or increases the number of connected components if the graph was already disconnected). Edge directions are ignored.

Highlight bridges in a network.

Find and highlight the source and sink vertices of a random acyclic graph.

```
g = DirectedGraph[RandomGraph[{10, 20}], "Acyclic",
VertexLabels -> "Name", VertexSize -> Large, EdgeStyle -> Gray]
```

Undirected graphs have neither source nor sink vertices because undirected edges are counted as bidirectional.

The exception is isolated vertices, which are counted both as sources and sinks.

These are merely convenience functions that can be implemented straightforwardly as

`IGGiantComponent`

is a convenience function that returns the largest weakly connected component of graph. If there are multiple components of largest size, there is no guarantee about which one would be returned. If this is a concern, use `WeaklyConnectedComponents`

or `WeaklyConnectedGraphComponents`

instead.

```
g = RandomGraph[{200, 200}];
HighlightGraph[
g,
IGGiantComponent[g]
] // IGLayoutFruchtermanReingold
```

`IGGiantComponent`

takes all standard graph options.

Size of the giant component of a random subgraph of a grid graph.

```
g = IGSquareLattice[{30, 30}, "Periodic" -> True];
Table[
{k, VertexCount@
IGGiantComponent@Subgraph[g, RandomSample[VertexList[g], k]]},
{k, 1, VertexCount[g], 1}
] // ListPlot
```

`IGTreeQ`

checks if a graph is a tree. An undirected tree is a connected graph with no cycles. A directed tree is similar, with its edges oriented either away from a root vertex (out-tree or arborescence) or towards a root vertex (in-tree or anti-arborescence).

By convention, the null graph is not a tree.

This is an out-tree.

It is not also an in-tree.

It becomes an in-tree if we reverse its edges.

This graph is neither an out-tree nor an in-tree.

However, it becomes a tree if we ignore edge directions.

`IGForestQ`

is a convenience function that tests if all connected components of a graph are trees.

This graph is not a tree, but it is a forest.

By convention, the null graph is not a tree, but it is a forest.

Use the second argument to test for forests of out-trees or in-trees. By default, directed graphs are checked to be out-forests.

`IGStrahlerNumber`

computes the Horton–Strahler index of each vertex in a rooted tree. The tree must be directed—this is how the root is encoded. The Horton–Strahler index of the tree itself is the index of the root, i.e. the largest returned index. This measure is also called *stream order*, as it was originally used to characterize river networks.

To get the Horton–Strahler number of the tree, find the maximal element.

`IGStrahlerNumber`

requires a directed (i.e. rooted) tree as input.

Orient undirected trees, effectively specifying a root vertex, before passing them to `IGStrahlerNumber`

.

IGTreelikeComponents finds the tree-like components of an undirected graph by repeatedly identifying and removing degree-1 vertices. Vertices in the tree-like components are not part of any undirected cycle, nor are they on a path connecting vertices that belong to a cycle.

```
g = RandomGraph[{100, 100}];
HighlightGraph[
g,
IGTreelikeComponents[g]
] // IGLayoutFruchtermanReingold
```

Highlight both the edges and vertices of tree-like components.

```
g = IGGiantComponent@RandomGraph[{50, 50}];
HighlightGraph[
g,
Join[
Union @@ (IncidenceList[g, #] &) /@ IGTreelikeComponents[g],
IGTreelikeComponents[g]
]
]
```

Remove tree-like components.

Vertices incident to multi-edges or loop-edges are not part of tree-like components.

Find the shortest set of paths connecting a set of points in the plane:

The edge weights are preserved in the result.

Compute the total path length.

Find a *maximum* spanning tree by negating the weights before running the algorithm.

`IGRandomSpanningTree`

samples the spanning trees (or forests) of a graph uniformly by performing a loop-erased random walk. Edge directions are ignored.

If a spanning forest of the entire graph is requested using `IGRandomSpanningTree[g]`

, then the vertex names and ordering are preserved. If a spanning tree of only a single component is requested using `IGRandomSpanningTree[{g, v}]`

, then this is not the case.

Highlight a few random spanning trees of the Petersen graph.

```
g = PetersenGraph[];
HighlightGraph[g, #, GraphHighlightStyle -> "Thick"] & /@
IGRandomSpanningTree[g, 9]
```

If the input is a multi-graph, each edge will be considered separately for the purpose of spanning tree calculations. Thus the following graph has not 3, but 5 different spanning trees. Two pairs of these are indistinguishable based on their adjacency matrix due to the indistinguishability of the two parallel `1 \[UndirectedEdge] 2`

edges. However, since all 5 spanning trees are generated with equal probability, two of the 3 adjacency matrices will appear twice as frequently as the third one.

Edge directions are ignored for the purpose of spanning tree calculation. Thus the result may not be an out-tree.

Create mazes by taking random spanning trees of grid graphs.

```
g = GridGraph[{10, 10}, GraphStyle -> "Web"];
HighlightGraph[g, IGRandomSpanningTree[g],
GraphHighlightStyle -> "DehighlightHide"]
```

```
g = GridGraph[{6, 6, 6}, VertexCoordinates -> Tuples[Range[6], {3}]];
HighlightGraph[g, IGRandomSpanningTree[g],
GraphHighlightStyle -> "DehighlightHide"]
```

Generate a random spanning tree of the component containing vertex `8`

.

`IGSpanningTreeCount`

computes the number of spanning trees of a graph using Kirchhoff’s theorem. Multigraphs and directed graphs are supported.

The number of spanning trees of a directed graph, rooted in any vertex.

The number of spanning trees rooted in vertex `1`

.

`IGSpanningTreeCount`

works on large graphs.

Edge multiplicities are taken into account. Thus the following graph has not 3, but 5 different spanning trees.

In a directed graph, a vertex \(d\) is said to *dominate* a vertex \(v\) if every path from the root to \(v\) passes through \(d\). We say that \(d\) is an *immediate dominator* of \(v\) if it does not dominate any other dominator of \(v\).

A dominator tree of a graph consists of the same vertices as the graph, and the children of a vertex are those other vertices which it immediately dominates.

Find the dominator tree of a directed graph.

Vertices that cannot be reached from the specified root are left isolated in the returned graph.

`IGDominatorTree`

accepts all standard `Graph`

options.

Directly find the immediate dominators of vertices in a graph.

The immediate dominator of a vertex is its parent in the dominator tree.

Neither the root, nor vertices unreachable from the root are included in the keys of the returned association.

A \(k\)-core of a graph is a maximal subgraph in which each vertex has degree at least \(k\). The coreness of a vertex is the highest order of \(k\)-cores that contain it.

By default, edge directions are ignored, and multi-edges are considered.

Use the second argument to consider only in- or out-degrees.

A matching of a graph is also known as an independent edge set.

`IGMaximumMatching`

ignores edge directions and edge weights.

The matching number of a graph is the size of its maximum matchings.

`IGUnfoldTree`

creates a tree based on the breadth-first traversal of a graph. Each time a graph vertex is found, a new tree vertex is created.

Available options:

`DirectedEdges -> False`

will ignore edge directions in directed graphs. Otherwise, the search is done only along edge directions.

The original vertex that generates a tree node is stored in the `"OriginalVertex"`

property.

We can label the tree nodes with the name of the original vertex either using pattern matching in `VertexLabels`

along with `PropertyValue`

…

```
IGLayoutReingoldTilford[
tree,
"RootVertices" -> {1},
VertexLabels -> (v_ :> PropertyValue[{tree, v}, "OriginalVertex"])
]
```

… or using `IGVertexMap`

.

```
IGLayoutReingoldTilford[tree, "RootVertices" -> {1}] //
IGVertexMap[# &, VertexLabels -> IGVertexProp["OriginalVertex"]]
```

In directed graphs, the search is done along edge directions. It may be necessary to give multiple starting roots to fully unfold a weakly connected (or unconnected) graph.

```
IGUnfoldTree[Graph[{1 -> 2, 2 -> 3}], {2, 1}] //
IGVertexMap[# &, VertexLabels -> IGVertexProp["OriginalVertex"]]
```

Use `DirectedEdges -> False`

to ignore edge directions during the search. Edge directions are still preserved in the result.

```
IGUnfoldTree[Graph[{1 -> 2, 2 -> 3}], {2}, DirectedEdges -> False] //
IGVertexMap[# &, VertexLabels -> IGVertexProp["OriginalVertex"]]
```

`IGNullGraphQ`

returns `True`

only for the null graph, i.e. the graph that has no vertices.

For graphs that have vertices, but no edges, it returns `False`

.

In contrast, the built-in `EmptyGraphQ`

tests if there are no edges:

`IGCompleteQ`

tests if a graph is complete, i.e. if all pairs of vertices are connected.

`IGCompleteQ`

ignores self-loops and multi-edges.

Check if each connected component of a graph is a clique.

The null graph is considered complete.

`IGCactusQ`

tests if a graph is a cactus. A cactus graph is a connected undirected graph in which any two simple cycles share at most one vertex. Equivalently, a cactus is a connected graph in which every edge belongs to at most one simple cycle.

`IGCactusQ`

supports multigraphs and ignores self-loops.

The null graph is not considered to be a cactus, but the singleton graph is.

Currently, `IGCactusQ`

does not support directed graphs.

IGraph/M’s motif-related functions count the number of times each possible connectivity pattern of \(k\) vertices (i.e. induced subgraph of size \(k\)) occurs in a graph. The patterns are called *motifs*. As of IGraph/M 0.4, only size 3 and 4 motifs are supported, and only (weakly) connected subgraphs are considered.

To count larger induced subgraphs, see `IGLADSubisomorphismCount`

. To identify where a subgraph occurs, see `IGLADFindSubisomorphisms`

.

To count non-connected size-3 subgraphs, use `IGTriadCensus`

.

igraph’s motif functions use the RAND-ESU algorithm, which is able to uniformly sample a random subset of motifs (connected subgraphs), and can thus estimate motif counts even in very large graphs. See the description of `IGMotifs`

for an example.

- S. Wernicke,
*Efficient Detection of Network Motifs*, IEEE/ACM Trans. Comput. Biol. Bioinforma.**3**, 347 (2006).

`IGMotifs`

counts how many times each motif (i.e. induced subgraph) of the given size occurs in the graph. For subgraphs that are not weakly connected, `Indeterminate`

is returned.

Available options are:

`DirectedEdges -> False`

treats the graph as undirected and`DirectedEdges -> True`

treats the graph as directed. The default is`DirectedEdges -> Automatic`

, which respects the directedness of the graph.

Motifs are returned by their `IGIsoclass`

, i.e. the same order as listed in `IGData`

.

Let us count size-3 motifs in the following graph, and summarize them a table. For non-weakly-connected subgraphs, `Indeterminate`

is returned.

Empty graphs are treated as undirected by default. To treat them as directed, use `DirectedEdges -> True`

. The result will be different as the number of non-isomorphic graphs on \(k\) vertices is not the same in the directed and undirected cases.

Let us find the size-4 motifs that stand out in the *E. coli* metabolic network by comparing the motif counts to that of a rewired graph:

`{Indeterminate, Indeterminate, Indeterminate, 1.36576, Indeterminate, \ Indeterminate, Indeterminate, 1.38326, 0.332702, Indeterminate, \ Indeterminate, Indeterminate, 0.593199, 0.641075, 0., Indeterminate, \ 0.0163017, 0., 0., 6.32195, 0., 0., Indeterminate, Indeterminate, \ 1.35664, 0.317772, 0.136506, Indeterminate, Indeterminate, 0.668807, \ 0., 0.0178394, 0., Indeterminate, Indeterminate, 0., 0., 0., 0., \ Indeterminate, 0.204245, 0.724535, 0., 0.205228, 0., 0.194249, \ 0.045118, 0., 0., 0., 0., 0., 1.51568, 0., 0.0740741, 0., 0., 0., 0., \ 0., 0., 0., Indeterminate, 0., 0., 0., 35.5991, 0., 0.202166, 0., 0., \ 0., 0., 0.126984, 0., 0., 1.30301, 0., 0., 0., 0., 0., 0., 0., 0., \ 0., 0., 0., 0., 0., 0., 0., 0.303743, 0.102826, 1.03087, 0., \ 0.020851, 0., 0.232549, 0., 0.0401769, 0., 0., 0., 0., 0., 0., 0., \ 0., 0., 0.670103, 0., 0.363636, 0., 0., 0., 0., 0., 0., 0., \ Indeterminate, 0.245413, 0., 0., 0., 1.64, 0., 0., 43.1355, 0., \ 0.585, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \ 1.94175, 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.142857, \ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., \ 0., 0., 0., 0., 0., 0., 0., 0., 0., Indeterminate, 0., 0., 0., 0., \ 0., 0., 0., 0., 0., Indeterminate, 0., 0., Indeterminate, \ Indeterminate}`

There are two motifs that are more than 30 times more common in the metabolic network than in the rewired graph.

The Davidson–Harel algorithm attempts to reduce edge crossings and can draw these subgraphs in a clearer way:

`IGMotifs`

uses the RAND-ESU algorithm which can uniformly sample a random subset of motifs, and thus estimate motif counts even in very large graphs. To enable random sampling, set a cutoff probability for stopping the search at each level of the ESU tree. The length of the cutoff probability vector, `n`

, must be the same as the motif size. The number of sampled motifs is, on average, a fraction of the total number.

Sample a fraction \(0.1^3=0.001\) of all motifs.

Sample 12.5% of motifs, i.e. a fraction of \(0.5^3\).

`IGMotifsVertexParticipation`

counts how many times each vertex participates in each motif. For each vertex, the result is returned in the same format as with `IGMotifs`

.

Available options are:

`DirectedEdges -> False`

treats the graph as undirected and`DirectedEdges -> True`

treats the graph as directed. The default is`DirectedEdges -> Automatic`

, which respects the directedness of the graph.

Count how many times each vertex appears in each 3-motif in a directed graph.

`<|"A" -> {Indeterminate, Indeterminate, 0, Indeterminate, 2, 0, 0, 2, 1, 1, 0, 2, 0, 0, 0, 0}, "B" -> {Indeterminate, Indeterminate, 0, Indeterminate, 1, 1, 0, 3, 0, 2, 0, 1, 1, 1, 0, 0}, "C" -> {Indeterminate, Indeterminate, 1, Indeterminate, 1, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0}, "D" -> {Indeterminate, Indeterminate, 0, Indeterminate, 2, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0}, "E" -> {Indeterminate, Indeterminate, 1, Indeterminate, 0, 0, 0, 2, 1, 2, 0, 1, 1, 0, 0, 0}, "F" -> {Indeterminate, Indeterminate, 1, Indeterminate, 3, 0, 0, 2, 0, 1, 0, 1, 0, 1, 0, 0}|>`

The sum of the participation counts in 3-motifs is 3 times the total motif counts of the graph.

`IGMotifsTotalCount[graph, motifSize]`

counts the number of weakly connected subgraphs of the given size in a graph.

`IGMotifsTotalCountEstimate[graph, motifSize, sampleSize]`

estimates the total number of motifs by taking a random subset of vertices of the specified size, and counting motifs in which these vertices participate. The total number is estimated as `motifCount*vertexCount/sampleSize`

. `IGMotifsTotalCountEstimate[graph, motifSize, vertices]`

uses the specified vertices as the sample.

Let us create a graph.

The number of size-4 subgraphs it has is:

However, only a small fraction of these is connected:

`IGMotifsTotalCount`

is effectively equivalent to (but much faster than) the following:

Estimate the count of connected subgraphs by subsampling: at each level of the ESU tree, continue only with probability 0.9.

Estimate the count of connected subgraphs by considering a random subset of 15 vertices (out of a total of 20).

Use the first 15 vertices tot estimate the count.

See `IGData["MANTriadLabels"]`

for the mapping between MAN labels and graphs.

`IGTriadCensus[g]`

does not return triad counts in the same order as `IGMotifs[g, 3]`

, i.e. ordered according to the triads’ `IGIsoclass[]`

. To get the result ordered by isoclass, use

`IGData["MANTriadLabels"]`

are ordered according to isoclass.

Highlight all triangles in a graph.

```
HighlightGraph[g, Subgraph[g, #], ImageSize -> Tiny,
GraphHighlightStyle -> "Thick"] & /@ IGTriangles[g]
```

Label a graph’s vertices based on the number of adjacent triangles.

```
RandomGraph[{8, 16}, VertexSize -> Large] //
IGVertexMap[Placed[#, Center] &,
VertexLabels -> IGAdjacentTriangleCount]
```

Triangle-free graphs do not have any fully connected subgraphs of size 3. Equivalently, they do not have any cliques (other than 2-cliques, which are edges).

Mycielski graphs are triangle-free.

igraph implements three isomorphism testing algorithms: BLISS, VF2 and LAD. These support slightly different functionality.

**Naming:** Most of IGraph/M’s isomorphism related functions include the name of the algorithm as a prefix, e.g. `IGBlissIsomorphicQ`

. Functions named as `…GetIsomorphism`

will find a single isomorphism. Functions named as `…FindIsomorphisms`

can find multiple isomorphisms. Both return a result in a format compatible with the built-in `FindGraphIsomorphism`

.

Additionally, `IGIsomorphicQ[]`

and `IGSubisomorphicQ[]`

try to select the best algorithm for the given graphs. For graphs without multi-edges, they use igraph’s default algorithm selection. For multigraphs, they use VF2 after internally transforming the multigraphs to edge- and vertex-coloured simple graphs, in a manner similar to `IGColoredSimpleGraph`

.

`IGIsomorphicQ`

decides if two graphs are isomorphic.

`IGIsomorphicQ`

supports multigraphs.

Get a specific mapping between the vertices of the graphs.

When the graphs are not isomorphic, an empty list is returned.

`IGSubisomorphicQ`

decides if a subgraph is part of a larger graph.

A dodecahedral graph does not contain a `[1, 2, 3]`

symmetric tree.

It does contain a `[3, 2, 1]`

tree.

Let us retrieve a specific mapping …

… and highlight it.

`IGSubisomorphicQ`

supports multigraphs.

The Bliss library was developed by Tommi Junttila and Petteri Kaski. It is capable of canonical labelling of directed or undirected vertex coloured graphs.

Bliss generally outperforms *Mathematica*’s built-in isomorphisms functions (including finding and counting automorphisms) as of *Mathematica* 12.0. However, this advantage will only be apparent for large and difficult graphs. For small ones the overhead of having to copy the graph and convert it to igraph’s internal format is much larger than the actual computation time.

All Bliss functions take a `"SplittingHeuristics"`

option, which can influence the performance of the method. Possible values are:

`"First"`

– First non-unit cell. Very fast but may result in large search spaces on difficult graphs. Use for large but easy graphs.`"FirstSmallest"`

– First smallest non-unit cell. Fast, should usually produce smaller search spaces than`"First"`

.`"FirstLargest"`

– First largest non-unit cell. Fast, should usually produce smaller search spaces than`"First"`

.`"FirstMaximallyConnected"`

– First maximally non-trivially connected non-unit cell. Not so fast, should usually produce smaller search spaces than`"First"`

,`"FirstSmallest"`

and`"FirstLargest"`

.`"FirstSmallestMaximallyConnected"`

– First smallest maximally non-trivially connected non-unit cell. Not so fast, should usually produce smaller search spaces than`"First"`

,`"FirstSmallest"`

and`"FirstLargest"`

.`"FirstLargestMaximallyConnected"`

– First largest maximally non-trivially connected non-unit cell. Not so fast, should usually produce smaller search spaces than`"First"`

,`"FirstSmallest"`

and`"FirstLargest"`

.

The default setting is `"FirstLargest"`

, which performs well on average on sparse graphs.

**Note:** The result of the `IGBlissCanonicalLabeling`

, `IGBlissCanonicalPermutation`

and `IGBlissanonicalGraph`

functions depend on the choice of `"SplittingHeuristics"`

. See the Bliss documentation for more information.

Let us take the cuboctahedral graph from GraphData …

… and also generate it based on its LCF notation.

The two graphs are isomorphic:

One particular mapping between them is the following:

How many mappings are there in total? The same number as the number of automorphisms of either graph.

Bliss cannot generate all 48 of these mappings *directly*. We can either use VF2 for this …

… or we can use the automorphism group computed by the `IGBlissAutomorphismGroup`

function.

Ask for all 48 vertex permutations that create isomorphic graphs:

`{{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, {1, 3, 2, 5, 4, 6, 7, 9, 8, 11, 10, 12}, {1, 4, 5, 2, 3, 7, 6, 10, 11, 8, 9, 12}, {1, 5, 4, 3, 2, 7, 6, 11, 10, 9, 8, 12}, {2, 1, 6, 4, 8, 3, 10, 5, 9, 7, 12, 11}, {2, 4, 8, 1, 6, 10, 3, 7, 12, 5, 9, 11}, {2, 6, 1, 8, 4, 3, 10, 9, 5, 12, 7, 11}, {2, 8, 4, 6, 1, 10, 3, 12, 7, 9, 5, 11}, {3, 1, 6, 5, 9, 2, 11, 4, 8, 7, 12, 10}, {3, 5, 9, 1, 6, 11, 2, 7, 12, 4, 8, 10}, {3, 6, 1, 9, 5, 2, 11, 8, 4, 12, 7, 10}, {3, 9, 5, 6, 1, 11, 2, 12, 7, 8, 4, 10}, {4, 1, 7, 2, 10, 5, 8, 3, 11, 6, 12, 9}, {4, 2, 10, 1, 7, 8, 5, 6, 12, 3, 11, 9}, {4, 7, 1, 10, 2, 5, 8, 11, 3, 12, 6, 9}, {4, 10, 2, 7, 1, 8, 5, 12, 6, 11, 3, 9}, {5, 1, 7, 3, 11, 4, 9, 2, 10, 6, 12, 8}, {5, 3, 11, 1, 7, 9, 4, 6, 12, 2, 10, 8}, {5, 7, 1, 11, 3, 4, 9, 10, 2, 12, 6, 8}, {5, 11, 3, 7, 1, 9, 4, 12, 6, 10, 2, 8}, {6, 2, 3, 8, 9, 1, 12, 4, 5, 10, 11, 7}, {6, 3, 2, 9, 8, 1, 12, 5, 4, 11, 10, 7}, {6, 8, 9, 2, 3, 12, 1, 10, 11, 4, 5, 7}, {6, 9, 8, 3, 2, 12, 1, 11, 10, 5, 4, 7}, {7, 4, 5, 10, 11, 1, 12, 2, 3, 8, 9, 6}, {7, 5, 4, 11, 10, 1, 12, 3, 2, 9, 8, 6}, {7, 10, 11, 4, 5, 12, 1, 8, 9, 2, 3, 6}, {7, 11, 10, 5, 4, 12, 1, 9, 8, 3, 2, 6}, {8, 2, 10, 6, 12, 4, 9, 1, 7, 3, 11, 5}, {8, 6, 12, 2, 10, 9, 4, 3, 11, 1, 7, 5}, {8, 10, 2, 12, 6, 4, 9, 7, 1, 11, 3, 5}, {8, 12, 6, 10, 2, 9, 4, 11, 3, 7, 1, 5}, {9, 3, 11, 6, 12, 5, 8, 1, 7, 2, 10, 4}, {9, 6, 12, 3, 11, 8, 5, 2, 10, 1, 7, 4}, {9, 11, 3, 12, 6, 5, 8, 7, 1, 10, 2, 4}, {9, 12, 6, 11, 3, 8, 5, 10, 2, 7, 1, 4}, {10, 4, 8, 7, 12, 2, 11, 1, 6, 5, 9, 3}, {10, 7, 12, 4, 8, 11, 2, 5, 9, 1, 6, 3}, {10, 8, 4, 12, 7, 2, 11, 6, 1, 9, 5, 3}, {10, 12, 7, 8, 4, 11, 2, 9, 5, 6, 1, 3}, {11, 5, 9, 7, 12, 3, 10, 1, 6, 4, 8, 2}, {11, 7, 12, 5, 9, 10, 3, 4, 8, 1, 6, 2}, {11, 9, 5, 12, 7, 3, 10, 6, 1, 8, 4, 2}, {11, 12, 7, 9, 5, 10, 3, 8, 4, 6, 1, 2}, {12, 8, 9, 10, 11, 6, 7, 2, 3, 4, 5, 1}, {12, 9, 8, 11, 10, 6, 7, 3, 2, 5, 4, 1}, {12, 10, 11, 8, 9, 7, 6, 4, 5, 2, 3, 1}, {12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}}`

Permuting the adjacency matrix with any of these leaves it invariant.

```
perms = PermutationList[#, VertexCount[g1]] & /@
GroupElements[group];
Equal @@ (AdjacencyMatrix[g1][[#, #]] & /@ perms)
```

Bliss works by computing a canonical labelling of vertices. Then isomorphism can be tested for by comparing the canonically relabelled graphs.

`IGBlissCanonicalGraph`

returns graphs in a consistent format so that two graphs are isomorphic if and only if their canonical graphs will compare equal with `===`

. Note that in *Mathematica*, graphs may not always compare equal even if they have the same vertex and edge lists.

The corresponding permutation and labelling are

Notice that the canonical labelling is simply

Also notice that it is a mapping from `g1`

to `IGBlissCanonicalGraph[g1]`

:

The canonical graph returned by `IGBlissCanonicalGraph`

always has vertices labelled by the integers `1, 2, …`

It can also be used to filter duplicates from a list of graphs

For example, let us generate all possible adjacency matrices of 3-vertex simple directed graphs.

```
(* fills nondiagonal entries of n by n matrix from vector *)
toMat[vec_, n_] :=
SparseArray@
Partition[Flatten@Riffle[Partition[vec, n], 0, {1, -1, 2}], n]
```

There are `2^(3 2) = 2^6 = 64`

such matrices.

```
graphs = AdjacencyGraph[toMat[#, 3], DirectedEdges -> True] & /@
IntegerDigits[Range[2^6] - 1, 2, 6];
```

But only 16 of them correspond to non-isomorphic graphs

When `IGBlissCanonicalGraph`

is given a vertex coloured graph, it will encode the colours into a vertex property named `"Color"`

. This allows distinguishing between graphs whose canonical graphs are identical in structure, but differ in colouring.

Take for example the following coloured graphs:

```
g = Graph[{1 \[UndirectedEdge] 2, 2 \[UndirectedEdge] 3},
VertexSize -> Large, GraphStyle -> "BasicBlack"];
colg1 = Graph[g,
Properties -> {1 -> {"color" -> 1}, 2 -> {"color" -> 3},
3 -> {"color" -> 2}}];
colg2 = Graph[g,
Properties -> {1 -> {"color" -> 1}, 2 -> {"color" -> 3},
3 -> {"color" -> 1}}];
```

Visualize them for clarity:

The vertex and edge lists of their canonical graphs are identical:

```
cang1 = IGBlissCanonicalGraph[{colg1, "VertexColors" -> "color"}];
cang2 = IGBlissCanonicalGraph[{colg2, "VertexColors" -> "color"}];
```

But they differ in colouring, and therefore do not compare equal:

The performance of Bliss functions may depend significantly on the choice of splitting heuristics.

```
g = LineGraph@GraphData[{"Hadamard", {24, 6}}];
timings = {#,
First@Timing@
IGBlissAutomorphismGroup[g,
"SplittingHeuristics" -> #]} & /@
{"First",
"FirstSmallest", "FirstLargest", "FirstMaximallyConnected",
"FirstSmallestMaximallyConnected",
"FirstLargestMaximallyConnected"};
TableForm[timings,
TableHeadings -> {None, {"Splitting heuristics", "Timing (s)"}}]
```

Let us visualize the vertex equivalence classes induced by a graph’s automorphism group. Two vertices are considered equivalent if there is an automorphism that maps one into the other.

```
With[{g = GraphData[{"Mycielski", 4}]},
HighlightGraph[g, GroupOrbits@IGBlissAutomorphismGroup[g],
VertexSize -> Large, GraphStyle -> "BasicBlack"]
]
```

Visualize the edge equivalence classes of a polyhedron, induced by its skeleton’s automorphism group.

```
With[{g = IGMeshGraph[mesh, VertexStyle -> Black]},
HighlightGraph[g,
EdgeList[g][[#]] & /@
GroupOrbits@IGBlissAutomorphismGroup@LineGraph[g]
]
]
```

- T. Junttila, P. Kaski, Engineering an Efficient Canonical Labeling Tool for Large and Sparse Graphs, 2007 Proceedings of the Ninth Workshop on Algorithm Engineering and Experiments, doi:10.1137/1.9781611972870.13.

VF2 supports vertex coloured and edge coloured graphs. A colour specification consists of one or more of the `"VertexColors"`

and `"EdgeColors"`

options. Allowed formats for these options are a list of integers, an association assigning integers to the vertices/edges, or `None`

. When using associations, it is not necessarily to specify a colour for each vertex/edge. The omitted ones are assumed to have colour `0`

.

The VF2 algorithm only supports simple graphs.

The following graph has two automorphisms: `{1, 2}`

and `{2, 1}`

.

If we colour one of the vertices, the permutation `{2, 1}`

becomes forbidden, so only one automorphism remains.

Multigraphs are not directly supported for isomorphism checking, but we can map the multigraph isomorphism problem into an edge-coloured graph isomorphism one by designating the multiplicity of each edge as its colour.

Since `g1`

and `g2`

are undirected, we need to bring their edges into a sorted canonical form before counting them. This ensures that `4 \[UndirectedEdge] 3`

and `3 \[UndirectedEdge] 4`

are treated as the same edge.

```
IGVF2IsomorphicQ[{Graph@Keys[colors1],
"EdgeColors" -> colors1}, {Graph@Keys[colors2],
"EdgeColors" -> colors2}]
```

`IGIsomorphicQ`

and `IGSubisomorphicQ`

check multigraph isomorphism in a similar way, based on edge colouring.

- L. P. Cordella, P. Foggia, C. Sansone, and M. Vento, IEEE Trans. Pattern Anal. Mach. Intell. 26, 1367 (2004).

The LAD library was developed by Christine Solnon. It is capable of finding subgraphs in a larger graph.

The LAD algorithm only supports simple graphs.

With the `"Induced" -> True`

option LAD will search for induced subgraphs.

Highlight subgraphs in a grid graph.

```
g = GridGraph[{3, 3}];
HighlightGraph[g, Subgraph[g, #], GraphHighlightStyle -> "Thick"] & /@
Union[Sort@*Values /@
IGLADFindSubisomorphisms[GridGraph[{2, 2}], g]]
```

Count how many times each vertex of a graph appears at the apex of the following subgraph (motif):

Generate a directed random graph to do the counting in.

`IGShorthand`

provides a concise way to input this subgraph.

This motif has a two-fold symmetry, as revealed by its automorphism group. We divide the final counts by two.

```
Counts@Lookup[
IGLADFindSubisomorphisms[motif, g, "Induced" -> True],
1
]/IGBlissAutomorphismCount[motif]
```

Check that a graph is claw-free.

```
clawFreeQ[graph_?UndirectedGraphQ] :=
Not@IGLADSubisomorphicQ[
StarGraph[4], (* claw graph *)
graph,
"Induced" -> True
]
```

- Christine Solnon,
*AllDifferent*-based filtering for subgraph isomorphism, Artificial Intelligence 174 (2010), doi:10.1016/j.artint.2010.05.002

All three included isomorphism algorithms support vertex coloured graphs, and VF2 supports edge coloured graphs as well. A coloured graph is specified as `{g, "VertexColors" -> …, "EdgeColors" -> …}`

, where both vertex and edge colour specifications are optional. Colours are represented by integers and may be specified in one of the following ways:

A list of integers, given in the same order as

`VertexList[g]`

(or`EdgeList[g]`

if specifying edge colours).`{Graph[{a, b}, {a \[UndirectedEdge] b}], "VertexColors" -> {1, 2}}`

.An association assigning integers to vertices (or edges). Vertices (or edges) not present in the association are assumed to have colour

`0`

.`{Graph[{a \[UndirectedEdge] b}], "VertexColors" -> <|a -> 1, b -> 2|>}`

.The name of a a vertex (or edge) property. Vertices (or edges) without an assigned property value are assumed to have colour

`0`

.`{Graph[{Property[a, "color" -> 1], Property[b, "color" -> 2]}, {a \[UndirectedEdge] b}], "VertexColors" -> "color"}`

`"VertexColors" -> None`

indicates no colouring.

**Example.** Define a graph along with the colours of its vertices.

Visualize it.

```
Graph[g,
VertexStyle -> Normal[ColorData[24] /@ vcols],
VertexSize -> Medium, VertexLabels -> Placed["Name", Center]
]
```

Compute its automorphism group, taking vertex colours into account.

The functions in this section test for properties related to a graph’s automorphism group. The summary table below illustrates the functions on a set of graphs which all have different properties.

```
graphs = {StarGraph[4], IGSquareLattice[{2, 3}, "Periodic" -> True],
HypercubeGraph[3], GraphData[{"Rook", {4, 4}}],
GraphData["ShrikhandeGraph"], GraphData["HoltGraph"],
GraphData["Tutte12Cage"], GraphData[{"Paulus", {25, 1}}]};
functions = <|
"regular" -> IGRegularQ,
"strongly regular" -> IGStronglyRegularQ,
"distance regular" -> IGDistanceRegularQ,
"vertex transitive" -> IGVertexTransitiveQ,
"edge transitive" -> IGEdgeTransitiveQ,
"arc transitive" -> IGEdgeTransitiveQ@*DirectedGraph,
"distance transitive" -> IGDistanceTransitiveQ
|>;
TableForm[
Through[Values[functions][#]] & /@ graphs,
TableHeadings -> {Show[#, ImageSize -> 50] & /@ graphs,
Keys[functions]},
TableDirections -> Row
] // Style[#, "Text"] &
```

`IGRegularQ`

checks if a graph is regular. All vertices of a regular graph have the same degrees. In regular directed graphs, the in- and out-degrees are also equal to each other.

Check if a graph is \(k\)-regular for \(k=2\) and \(k=3\).

The null graph is considered 0-regular.

Check if a directed graph is regular.

`IGRegularQ`

considers self-loops and multi-edges when computing vertex degrees.

`IGStronglyRegularQ`

checks if a graph is strongly regular. A strongly regular graph is a regular graph where each pair of connected vertices have the same number of common neighbours, \(\lambda\), and each pair of unconnected vertices also have the same number of common neighbours, \(\mu\).

Hypercube graphs and 3 and higher dimensions are not strongly regular, even though they are regular.

Some authors exclude empty and complete graph from the definition, as they satisfy these conditions trivially. `IGStronglyRegularQ`

returns `True`

for these.

It also returns `True`

for graphs on 0, 1 and 2 vertices.

Currently, `IGStronglyRegularQ`

does not support directed graphs.

`IGStronglyRegularParameters`

returns the parameters \((v,k,\lambda ,\mu )\) of a strongly regular graph. \(v\) is the number of vertices, \(k\) the degree of the vertices, \(\lambda\) the number of common neighbours of connected vertices and \(\mu\) the number of common neighbours of unconnected vertices.

The parameters of a strongly regular graph satisfy the equation \((v-k-1)\mu =k(k-\lambda -1)\).

\(\lambda\) and \(\mu\) are not well-defined for empty and complete graphs, respectively. In these cases, 0 is returned.

For non-strongly-regular graphs, `{}`

is returned.

`IGDistanceRegularGraph`

checks if a graph is distance regular.

A distance regular graph with a diameter of 2 is also strongly regular.

The Shrikhande graph is the smallest graph that is distance regular, but not distance transitive.

A disconnected graph is distance regular if its components are distance regular and they are co-spectral. The following graphs are co-spectral:

They are both distance regular with the same intersection array.

Thus their disjoint union is also distance regular.

All distance transitive graphs are also distance regular, but the reverse is not true.

`IGDistanceRegularQ`

does not currently support directed graphs or non-simple graphs.

For non-distance-regular graphs, `{}`

is returned.

`IGIntersectionArray`

does not currently support directed graphs.

`IGVertexTransitiveQ`

checks if a graph is vertex transitive, i.e. if any vertex can be mapped into any other by some automorphism of the graph.

All Cayley graphs are vertex transitive.

`IGEdgeTransitiveQ`

checks if a graph is edge transitive, i.e. if any edge can be mapped into any other by some automorphism of the graph.

The Folkman graph is not vertex transitive but it is edge transitive.

`IGEdgeTransitiveQ`

takes into account edge directions.

Arc transitivity in an undirected graph refers to edge transitivity when each undirected edge is replaced by two opposite directed edges.

Some graphs are edge transitive, but not arc transitive.

Most graphs are edge transitive if their line graphs are vertex transitive. The exceptions are disjoint unions of the 3-star and 3-cycle. These two graphs have the same line graph, but they are not isomorphic.

`IGSymmetricQ`

checks if a graph is both vertex transitive and edge transitive. Note that this property is distinct from being *arc transitive*, which is the definition used for “s*ymmetric”* by some authors.

Make a table of symmetric graphs up to size 7:

```
Grid[
Table[
Graph[#, ImageSize -> 50, PlotTheme -> "Business"] & /@
Select[GraphData /@ GraphData[k], IGSymmetricQ],
{k, 1, 7}
], Frame -> All, ItemSize -> All]
```

Some authors use the term *symmetric graph* to refer to arc transitive graphs. Arc transitivity can be checked using `IGEdgeTransitiveQ@DirectedGraph[#] &`

. All arc-transitive graphs are both vertex- and edge-transitive, but the reverse is not true. The smallest graph that is both vertex- and edge-transitive, but not arc-transitive, is the 27-vertex Doyle graph, also known as the Holt graph.

`IGDistanceTransitiveQ`

checks if a graph is distance transitive. In a distance transitive graph, any two ordered pairs of vertices which are the same distance apart can be mapped into each other by some automorphism.

All Platonic graphs are distance transitive.

Some graphs are symmetric, but not distance transitive.

`IGDistanceTransitiveQ`

does not exclude non-connected graphs.

`IGDistanceTransitiveQ`

works with directed graphs.

```
g = With[{n = 11},
RelationGraph[
MemberQ[Rest@Union@Mod[Range[n]^2, n], Mod[#1 - #2, n]] &,
Range[n] - 1]
]
```

The following directed graph is vertex transitive, but not distance transitive.

`IGHomeomorphicQ`

tests if two graphs are homeomorphic, i.e. whether they have the same topological structure. Two graphs \(G_1\) and \(G_2\) are homeomorphic if there is an isomorphism from a subdivision of \(G_1\) to a subdivision of \(G_2\).

is effectively implemented as .

The following graphs are homeomorphic.

They smoothen to the same graph.

Any two cycle graphs are homeomorphic.

A cycle and a path graph are not homeomorphic.

A triangular and a square lattice on the same number of vertices are, in general, topologically different.

When testing empirical graphs for equivalence, it is often useful to remove tree-like components. For example, the face-face and the face-edge adjacency graphs of a geometric mesh are equivalent, save for the tree-like components.

A graph is called self-complementary if it is isomorphic with its complement.

The 4-vertex path graph is self-complementary.

Find all 3-vertex self-complementary directed graphs.

`IGColoredSimpleGraph`

is a helper function that encodes a non-simple graph (i.e a graph with self-loops or multi-edges) into an edge- and vertex-colored simple graph. The coloured simple graph can be used directly as an input to coloured isomorphism checking functions such as `IGVF2IsomorphicQ`

.

The vertex colours are computed as the multiplicity of self-loops at each vertex. The edge colours are computed as the multiplicities or non-loop edges.

The following graphs are not simple and cannot be used with `IGVF2IsomorphicQ`

directly.

`IGColoredSimpleGraph`

can encode them as coloured graphs. Its output can be supplied directly to `IGVF2IsomorphicQ`

.

Now can can determine that `g1`

is isomorphic to `g2`

, but not to `g3`

.

When searching for subgraphs in multigraphs with this method, be aware that a match occurs only if the edge multiplicities are the same. This sort of matching is useful e.g. in substructure search chemistry, where a double bond must only match another double bond, but not a single one.

Use `IGSubisomorphicQ`

to match any subgraph.

`IGMaximumFlowValue`

is equivalent to `IGMinimumCutValue`

except that it uses the `EdgeCapacity`

property instead of `EdgeWeight`

.

Edge capacities are taken from the `EdgeCapacity`

property.

Element \(F_{ij}\) of the flow matrix is the flow through the edge connecting the \(i\)th node to the \(j\)th one. In an undirected graph, \(F_{ij}=-F_{ij}\).

Edge capacities are taken from the `EdgeCapacity`

property.

Let us take a directed graph with edge capacities set …

… and compute the maximum flow between two of its vertices.

The result is returned as a sparse matrix containing the flows through each edge.

If the input is an undirected graph, the flow matrix contains entries of opposing sign for the two directions along each edge.

Unlike `IGEdgeConnectivity`

, `IGMinimumCutValue`

takes weights into account.

The minimum cut value of the null graph and singleton graph are returned as `0`

and `∞`

, respectively.

The Gomory–Hu tree is a weighted tree that encodes the minimum cuts between all pairs of vertices of an undirected graph. The Gomory–Hu tree has the same vertices as the graph it characterizes. The minimum cut between an \(s\)-\(t\) pair of the graph has the same size as smallest edge weight on the path from \(s\) to \(t\) in the Gomory–Hu tree.

Weighted graphs are supported.

The path from 1 to 9 is `1 \[UndirectedEdge] 2 \[UndirectedEdge] 5 \[UndirectedEdge] 9`

and has the weights `{3, 5, 4}`

. The smallest one, 3, is the minimum value of a cut separating 1 from 9.

The following examples are based on the ones in the R/igraph documentation.

This is the network from the Moody-White paper:

- J. Moody and D. R. White. Structural cohesion and embeddedness: A hierarchical concept of social groups. American Sociological Review, 68(1):103–127, Feb 2003.

```
mw = Graph[{"1" \[UndirectedEdge] "2", "1" \[UndirectedEdge] "3",
"1" \[UndirectedEdge] "4", "1" \[UndirectedEdge] "5",
"1" \[UndirectedEdge] "6", "2" \[UndirectedEdge] "3",
"2" \[UndirectedEdge] "4", "2" \[UndirectedEdge] "5",
"2" \[UndirectedEdge] "7", "3" \[UndirectedEdge] "4",
"3" \[UndirectedEdge] "6", "3" \[UndirectedEdge] "7",
"4" \[UndirectedEdge] "5", "4" \[UndirectedEdge] "6",
"4" \[UndirectedEdge] "7", "5" \[UndirectedEdge] "6",
"5" \[UndirectedEdge] "7", "5" \[UndirectedEdge] "21",
"6" \[UndirectedEdge] "7", "7" \[UndirectedEdge] "8",
"7" \[UndirectedEdge] "11", "7" \[UndirectedEdge] "14",
"7" \[UndirectedEdge] "19", "8" \[UndirectedEdge] "9",
"8" \[UndirectedEdge] "11", "8" \[UndirectedEdge] "14",
"9" \[UndirectedEdge] "10", "10" \[UndirectedEdge] "12",
"10" \[UndirectedEdge] "13", "11" \[UndirectedEdge] "12",
"11" \[UndirectedEdge] "14", "12" \[UndirectedEdge] "16",
"13" \[UndirectedEdge] "16", "14" \[UndirectedEdge] "15",
"15" \[UndirectedEdge] "16", "17" \[UndirectedEdge] "18",
"17" \[UndirectedEdge] "19", "17" \[UndirectedEdge] "20",
"18" \[UndirectedEdge] "20", "18" \[UndirectedEdge] "21",
"19" \[UndirectedEdge] "20", "19" \[UndirectedEdge] "22",
"19" \[UndirectedEdge] "23", "20" \[UndirectedEdge] "21",
"21" \[UndirectedEdge] "22", "21" \[UndirectedEdge] "23",
"22" \[UndirectedEdge] "23"}, VertexLabels -> "Name"];
```

`{{{"1", "2", "3", "4", "5", "6", "7", "21", "8", "11", "14", "19", "9", "10", "12", "13", "16", "15", "17", "18", "20", "22", "23"}, {"1", "2", "3", "4", "5", "6", "7", "21", "19", "17", "18", "20", "22", "23"}, {"7", "8", "11", "14", "9", "10", "12", "13", "16", "15"}, {"1", "2", "3", "4", "5", "6", "7"}, {"7", "8", "11", "14"}}, {1, 2, 2, 5, 3}}`

```
CommunityGraphPlot[mw, Rest@blocks,
CommunityRegionStyle ->
Table[Directive[Opacity[0.5], ColorData[96][i]], {i,
Length[blocks] - 1}]]
```

Science camp network:

```
sc = Graph[{"Pauline" \[UndirectedEdge] "Jennie",
"Pauline" \[UndirectedEdge] "Ann",
"Jennie" \[UndirectedEdge] "Ann",
"Jennie" \[UndirectedEdge] "Michael",
"Michael" \[UndirectedEdge] "Ann",
"Holly" \[UndirectedEdge] "Jennie",
"Jennie" \[UndirectedEdge] "Lee",
"Michael" \[UndirectedEdge] "Lee",
"Harry" \[UndirectedEdge] "Bert", "Harry" \[UndirectedEdge] "Don",
"Don" \[UndirectedEdge] "Bert", "Gery" \[UndirectedEdge] "Russ",
"Russ" \[UndirectedEdge] "Bert",
"Michael" \[UndirectedEdge] "John",
"Gery" \[UndirectedEdge] "John", "Russ" \[UndirectedEdge] "John",
"Holly" \[UndirectedEdge] "Pam", "Pam" \[UndirectedEdge] "Carol",
"Holly" \[UndirectedEdge] "Carol",
"Holly" \[UndirectedEdge] "Bill",
"Bill" \[UndirectedEdge] "Pauline",
"Bill" \[UndirectedEdge] "Michael",
"Bill" \[UndirectedEdge] "Lee", "Harry" \[UndirectedEdge] "Steve",
"Steve" \[UndirectedEdge] "Don",
"Steve" \[UndirectedEdge] "Bert",
"Gery" \[UndirectedEdge] "Steve",
"Russ" \[UndirectedEdge] "Steve",
"Pam" \[UndirectedEdge] "Brazey",
"Brazey" \[UndirectedEdge] "Carol", "Pam" \[UndirectedEdge] "Pat",
"Brazey" \[UndirectedEdge] "Pat",
"Carol" \[UndirectedEdge] "Pat", "Holly" \[UndirectedEdge] "Pat",
"Gery" \[UndirectedEdge] "Pat"}, VertexLabels -> "Name"];
```

`{{{"Pauline", "Jennie", "Ann", "Michael", "Holly", "Lee", "Harry", "Bert", "Don", "Gery", "Russ", "John", "Pam", "Carol", "Bill", "Steve", "Brazey", "Pat"}, {"Harry", "Bert", "Don", "Steve"}, {"Holly", "Pam", "Carol", "Brazey", "Pat"}, {"Pauline", "Jennie", "Ann", "Michael", "Lee", "Bill"}}, {2, 3, 3, 3}}`

A clique is a fully connected subgraph. An independent vertex set is a subset of a graph’s vertices with no connections between them.

*Mathematica*’s `FindClique`

function only finds maximal cliques. IGraph/M provides functions for finding or counting all cliques, i.e. complete subgraphs, of a graph.

Simply counting cliques is much more memory efficient (and faster) than returning all of them.

The clique finder in IGraph/M ignores edge directions.

To find cliques in directed graphs, convert them to undirected and keep mutual (bidirectional) edges only.

A clique cover of a graph is a partitioning of its vertices such that each partition forms a clique. `IGCliqueCover`

finds a minimum clique cover, i.e. a partitioning into a smallest number of cliques.

The clique cover number of a graph is the smallest number of cliques that can be used to cover its vertices.

Available `Method`

option values are:

`"Minimum"`

finds a minimum clique cover.`"Heuristic"`

is much faster, but the result is not typically a minimum cover.

Compute a minimum clique cover of a random graph.

Visualize the clique cover.

Find the clique cover number without returning a cover.

The clique cover problem is equivalent to the colouring of the complement graph. `IGCliqueCover`

is effectively implemented as

For difficult problems, it may be useful to use `IGMinimumVertexColoring`

or `IGVertexColoring`

directly instead of `IGCliqueCover`

, and tune their options to achieve better performance. See the `"ForcedColoring"`

option of `IGMinimumVertexColoring`

on how to do this.

The maximal cliques of the graph can approximate the scenes in which characters appear together.

We can construct a bipartite graph of connections between potential scenes and characters

```
IGLayoutBipartite[
Graph@Catenate[
Thread /@ Thread[Range@Length[cliques] <-> cliques]],
VertexSize -> 0.5, ImageSize -> 220
] // IGVertexMap[Placed[#, If[IntegerQ[#], Before, After]] &,
VertexLabels -> VertexList]
```

*Note:**The term “graphlet” is used for multiple unrelated concepts in the literature. This section deals with decomposing weighted graphs into cliques. If you are looking to count induced subgraphs, see the* `IGMotifs`

*function.*

```
g = IGShorthand["A,B,D,E,C, A-B-C-A, C-E-D-B, D-C, E-B",
EdgeWeight -> {2, 3, 2, 4, 4, 1, 4, 1},
EdgeLabels -> "EdgeWeight", VertexLabels -> None,
VertexShapeFunction -> "Name", PerformanceGoal -> "Quality",
GraphLayout -> "CircularEmbedding"
]
```

- Hossein Azari Soufiani and Edoardo M Airoldi, Graphlet decomposition of a weighted network, https://arxiv.org/abs/1203.2821

The following functions are available:

If you are looking for the Sugiyama layout from igraph, try the built-in `GraphLayout -> "LayeredDigraphEmbedding"`

, or `LayeredGraphPlot`

. These are also based on the Sugiyama algorithm.

Layout functions also take any standard `Graph`

option.

Many layout algorithms take the following options:

`"MaxIterations"`

controls either the *maximum* number of iterations performed by the algorithm or the *exact* number of iterations, depending on the specific algorithm and settings. The option name is the same for all functions to make it easier to interchange them when visualizing dynamic graphs.

`"Align" -> True`

aligns the output horizontally. Examples:

```
{IGLayoutFruchtermanReingold[IGSquareLattice[{2, 4}](*,
"Align" -> True is the default *)],
IGLayoutFruchtermanReingold[IGSquareLattice[{2, 4}],
"Align" -> False]}
```

`"Continue" -> True`

allows using existing vertex coordinates as starting points for algorithms that update vertex positions incrementally. We can use this to visualize how the layout algorithms work …

```
g = IGLayoutRandom@
RandomGraph[BarabasiAlbertGraphDistribution[100, 1]];
ListAnimate@
NestList[IGLayoutGraphOpt[#, "Continue" -> True,
"MaxIterations" -> 80] &, g, 40]
```

… or to visualize dynamic graph processes such as adding edges to the graph one by one:

```
g = IGLayoutKamadaKawai@
Graph[Range[25], {1 \[UndirectedEdge] 25},
VertexLabels -> "Name"];
ListAnimate@NestList[
IGLayoutKamadaKawai[
EdgeAdd[#, UndirectedEdge @@ RandomSample[VertexList[#], 2]],
"MaxIterations" -> 15, "Continue" -> True, "Align" -> False] &,
g,
30
]
```

Visualize a planar graph without edge crossings using the Davidson–Harel simulated annealing method, and taking starting coordinates from `GraphLayout -> "PlanarEmbedding"`

.

This layout avoids crossings, but it is not pleasing:

We can post process it while avoiding the introduction of any edge crossings:

```
IGLayoutDavidsonHarel[
IGVertexMap[# &,
VertexCoordinates -> (Rescale@
GraphEmbedding[#, "PlanarEmbedding"] &), g],
"Continue" -> True, "EdgeCrossingWeight" -> 1000
]
```

Several of the graph layout algorithms in igraph can take edge weights into accounts. How the weights are used during layout differs between them.

`IGLayoutFruchtermanReingold`

multiplies the attraction between vertices by the weights. Thus higher weights result in shorter edges.`IGLayoutKamadaKawai`

produces longer edges for higher weights

`IGLayoutReingoldTilford[]`

and `IGLayoutReingoldTilfordCircular[]`

are designed for laying out trees. The following options are available:

`"RootVertices"`

allows nodes to be used as the root nodes in the layout. It must be a list, even if there is a single root node. Multiple root nodes are meant to be used with forests.`DirectedEdges -> True`

lays out the graph so that directed edges are pointing form lower levels (near the root) towards higher ones (away from the root).`"Rotation"`

controls the orientation of the layout. It must be given in radians.

`IGLayoutBipartite`

draws a bipartite graph, attempting to minimize the number of edge crossing using the Sugiyama algorithm.

The available options are:

`"Orientation"`

can be`Horizontal`

or`Vertical`

`"PartitionGap"`

controls the size of the gap between the two partitions`"VertexGap"`

controls the minimum size of the gap between vertices in a partition`MaxIterations`

controls the maximum number of iterations performed during edge crossing minimization.`"BipartitePartitions"`

can be used to explicitly specify the partitioning of the graph.

By default, a partitioning is computed automatically.

```
g = Graph[{1 \[UndirectedEdge] 2, 3 \[UndirectedEdge] 4},
VertexLabels -> "Name"];
IGLayoutBipartite[g]
```

The partitioning can also be specified explicitly.

Draw a bipartite layout with curved edges.

`IGLayoutDrL`

is designed specifically for visualizing large graphs with high clustering. The following image is created using DrL and shows a `36000`

node network of collaborations between condensed matter scientists.

The image was generated using the following code:

```
lg = ExampleData[{"NetworkGraph",
"CondensedMatterCollaborations2005"}];
lg = IndexGraph@Subgraph[lg, First@ConnectedComponents[lg]];
c = IGCommunitiesMultilevel[lg]
pts = GraphEmbedding@
IGLayoutDrL[lg]; (* this takes a while *)
figure = Graphics[
GraphicsComplex[pts,
{
{White, AbsoluteThickness[0.3], Opacity[0.05],
Line[List @@@ EdgeList[lg]]},
{AbsolutePointSize[2], Opacity[0.7],
MapIndexed[
{ColorData[45]@First[#2], Point[#1]} &,
c["Communities"]
]}
}
],
Background -> Black
]
```

`shift`-`enter` evaluation is disabled in the cell above to avoid running it accidentally. Running the code takes about 2-3 minutes on a modern computer. Copy the code to a new cell to try it.

Create galleries of the various graph layouts available in IGraph/M.

Visualise a tree graph with all layouts.

```
g = RandomGraph[BarabasiAlbertGraphDistribution[30, 1]];
layouts =
Graph[#[g], PlotLabel -> #, LabelStyle -> 7] & /@ {IGLayoutCircle,
IGLayoutDavidsonHarel, IGLayoutDrL, IGLayoutDrL3D,
IGLayoutFruchtermanReingold, IGLayoutFruchtermanReingold3D,
IGLayoutGEM, IGLayoutGraphOpt, IGLayoutKamadaKawai,
IGLayoutKamadaKawai3D, IGLayoutRandom, IGLayoutReingoldTilford,
IGLayoutReingoldTilfordCircular, IGLayoutSphere,
IGLayoutBipartite, IGLayoutPlanar};
Multicolumn[layouts]
```

Visualise a polyhedral graph with all layouts.

```
g = GraphData["DodecahedralGraph"];
layouts =
Graph[#[g], PlotLabel -> #, LabelStyle -> 7] & /@ {IGLayoutCircle,
IGLayoutDavidsonHarel, IGLayoutDrL, IGLayoutDrL3D,
IGLayoutFruchtermanReingold, IGLayoutFruchtermanReingold3D,
IGLayoutGEM, IGLayoutGraphOpt, IGLayoutKamadaKawai,
IGLayoutKamadaKawai3D, IGLayoutRandom, IGLayoutSphere,
IGLayoutPlanar, IGLayoutTutte};
Multicolumn[layouts]
```

The following functions are available:

*Modularity* is defined for a given partitioning of a graph’s vertices into *communities*. It is defined as

\[Q=\frac{1}{2m}\sum _{i,j} \left(A_{ij}-\frac{k_ik_j}{2m}\right)\delta _{c_ic_j},\]

where \(m\) is the number of edges, \(A\) is the adjacency matrix, \(k_i\) is the degree of node \(i\), and \(c_i\) is the community that node \(i\) belongs to. \(\delta _{ij}\) is the Kronecker \(\delta\) symbol. For weighted graphs, \(A\) is the weighted adjacency matrix, \(k_i\) are the sum of weights of edges incident on node \(i\), and \(m\) is the sum of all weights.

Modularity characterizes the tendency of vertices to connect more within their own group than with other groups. For a given partitioning, it can be computed using `IGModularity`

. Community detection functions find a partitioning of the graph which results in high modularity.

Community detection functions return `IGClusterData`

objects.

The data available in the object can be queried using `IGClusterData[…]["Properties"]`

. See the Examples section below for more information.

`IGClusterData`

represents a partitioning of a graph into communities. This object cannot be created directly. It is returned by community detection functions. See the Examples section below for more information.

Query the available properties.

Retrieve the communities.

When the `"Modularity"`

property is available, `Max[cl["Modularity"]]`

gives the modularity of the current partitioning.

`IGModularity[graph, communities]`

is equivalent to `GraphAssortativity[graph, communities, "Normalized" -> False]`

.

`IGCommunitiesEdgeBetweenness[]`

implements the Girvan–Newman algorithm.

Weighted graphs are supported. Weights are treated as “distances”, i.e. a large weight represents a weak connection.

Available option values:

`"ClusterCount"`

, the number of communities to return. Default:`Automatic`

.

Special properties returned with the result:

`"RemovedEdges"`

is the list of edges removed in each step of the algorithm.`"Bridges"`

records the steps which resulted in splitting the graph into more components.

- M. Girvan and M. E. J. Newman: Community structure in social and biological networks,
*PNAS*99, 7821-7826 (2002).

`IGCommunitiesFluid[]`

implements the fluid communities algorithm.

- F. Parés, D. Garcia-Gasulla, A. Vilalta, J. Moreno, E. Ayguadé, Jesús Labarta, U. Cortés, T. Suzumura: Fluid Communities: A Competitive, Scalable and Diverse Community Detection Algorithm, https://arxiv.org/abs/1703.09307

`IGCommunitiesGreedy[]`

implements greedy optimization of modularity.

Weighted graphs are supported.

- A. Clauset, M. E. J. Newman, C. Moore: Finding community structure in very large networks, http://www.arxiv.org/abs/cond-mat/0408187

`IGCommunitiesInfoMAP[]`

implements the InfoMAP algorithm.

It supports both edge weights and vertex weights.

The default number of trials is 10.

M. Rosvall and C. T. Bergstrom, Maps of information flow reveal community structure in complex networks,

*PNAS*105, 1118 (2008)M. Rosvall, D. Axelsson, and C. T. Bergstrom, The map equation,

*Eur. Phys. J. Special Topics*178, 13 (2009)

Weighted graphs are supported.

- Raghavan, U.N. and Albert, R. and Kumara, S.: Near linear time algorithm to detect community structures in large-scale networks.
*Phys. Rev. E*76, 036106. (2007).

Weighted graphs are supported.

Available option values:

`"ClusterCount"`

, the number of communities to return. May return fewer communities than requested. Default:`Automatic`

.

- M. E. J. Newman: Finding community structure using the eigenvectors of matrices,
*Phys. Rev. E*74:036104 (2006).

`IGCommunitiesMultilevel[]`

implements the Louvain community detection method.

Weighted graphs are supported.

- V. D. Blondel, J.-L. Guillaume, R. Lambiotte and E. Lefebvre: Fast unfolding of community hierarchies in large networks,
*J. Stat. Mech.*P10008 (2008)

The Leiden algorithm is similar to the multilevel algorithm, often called the Louvain algorithm, but it is faster and yields higher quality solutions. It can optimize both modularity and the Constant Potts Model, which does not suffer from the resolution-limit (see preprint http://arxiv.org/abs/1104.3083).

The Leiden algorithm consists of three phases: (1) local moving of nodes, (2) refinement of the partition and (3) aggregation of the network based on the refined partition, using the non-refined partition to create an initial partition for the aggregate network. In the local move procedure in the Leiden algorithm, only nodes whose neighborhood has changed are visited. The refinement is done by restarting from a singleton partition within each cluster and gradually merging the subclusters. When aggregating, a single cluster may then be represented by several nodes (which are the subclusters identified in the refinement).

The Leiden algorithm provides several guarantees. The Leiden algorithm is typically iterated: the output of one iteration is used as the input for the next iteration. At each iteration all clusters are guaranteed to be connected and well-separated. After an iteration in which nothing has changed, all nodes and some parts are guaranteed to be locally optimally assigned. Finally, asymptotically, all subsets of all clusters are guaranteed to be locally optimally assigned.

The Leiden method maximizes a quality measure (a generalization of modularity) defined as

\[Q=\frac{1}{2m}\sum _{i,j}\left(A_{ij}-\gamma n_in_j\right)\delta _{c_ic_j}\]

where \(m\) is the sum of edge weights (number of edges if the graph is unweighted), \(A\) is the weighted adjacency matrix, \(n_i\) is the weight of vertex \(i\), and \(c_i\) is the community that vertex \(i\) belongs to. \(\delta _{ij}\) is the Kronecker \(\delta\) symbol.

\(\gamma\) is a resolution parameter that can be set with the `"Resolution"`

option.

The function chooses the vertex weights automatically, according to the value of the `VertexWeight`

option:

`VertexWeight -> "NormalizedStrength"`

(default) sets \(n_i=k_i/\sqrt{2m}\), where \(k_i\) is the strength (sum of incident edge weights) of vertex \(i\). If \(\gamma =1\), then the quality measure becomes equivalent to the modularity.`VertexWeight -> "Constant"`

sets \(n_i=1\). With this choice, it is recommended to set the resolution parameter \(\gamma\) explicitly. A reasonable \(\gamma\) value for unweighted graphs is the graph density.`VertexWeight -> "VertexWeight"`

takes vertex weights from the`VertexWeight`

graph property.

Other available options:

`"Resolution" -> γ`

sets the resolution parameter \(\gamma\). The default is \(\gamma =1\). With`VertexWeight -> "NormalizedStrength"`

, a reasonable value is 1. With`VertexWeight -> "Constant"`

, a reasonable value is the graph density.`"Beta" -> β`

sets the randomness used in the refinement step when merging clusters. The default is \(\beta =0.01\).

Special properties returned with the result :

`"Quality"`

is the value of the quality measure \(Q\).

Examples:

```
g = Graph[ExampleData[{"NetworkGraph", "LesMiserables"}],
GraphStyle -> "BasicBlack", VertexSize -> 2];
```

With the default option values `VertexWeight -> "NormalizedStrength"`

and `"Resolution" -> 1`

, `IGCommunitiesLeiden`

effectively uses the modularity as the quality measure.

A higher `"Resolution"`

value results in more communities.

With `VertexWeight -> "Constant"`

, it is recommended to set `"Resolution"`

explicitly. A reasonable starting point is `GraphDensity[g]`

.

```
HighlightGraph[
g,
IGCommunitiesLeiden[g, VertexWeight -> "Constant",
"Resolution" -> 0.1]["Communities"]
]
```

- Traag, V. A., Waltman, L., van Eck, N. J. (2019). From Louvain to Leiden: guaranteeing well-connected communities. Scientific Reports, 9(1), 5233. http://dx.doi.org/10.1038/s41598-019-41695-z

Finds the clustering that maximizes modularity exactly. This algorithm is very slow.

Weighted graphs are supported.

Weighted graphs are supported.

Option values for `Method`

are:

`"Original"`

only supports positive edge weights, but doesn’t check that the supplied weights are actually positive.`"Negative"`

supports negative weights as well.`Automatic`

selects`"Negative"`

if negative weights are presents and`"Original"`

otherwise.

Option values for `"UpdateRule"`

are: `"Simple"`

, `"Configuration"`

For

`Method -> "Original`

, see Joerg Reichardt and Stefan Bornholdt: Statistical Mechanics of Community Detection, http://arxiv.org/abs/cond-mat/0603718For

`Method -> "Negative"`

, see V. A. Traag and Jeroen Bruggeman: Community detection in networks with positive and negative links, http://arxiv.org/abs/0811.2329

`IGCommunitiesWalktrap[]`

finds communities using short random walks, exploiting the fact that random walks tend to stay within the same cluster.

Weighted graphs are supported.

The default number of steps is 4.

Available option values:

`"ClusterCount"`

, the number of communities to return. Default:`Automatic`

.

- Pascal Pons, Matthieu Latapy: Computing communities in large networks using random walks, http://arxiv.org/abs/physics/0512106

Community detection functions return `IGClusterData`

objects.

Various properties of these objects can be queried:

`{{"Myriel", "Napoleon", "Mlle Baptistine", "Mme. Magloire", "Countess De Lo", "Geborand", "Champtercier", "Cravatte", "Count", "Old Man"}, {"Labarre", "Valjean", "Marguerite", "Mme. De R", "Isabeau", "Gervais", "Bamatabois", "Perpetue", "Simplice", "Scaufflaire", "Woman1", "Judge", "Champmathieu", "Brevet", "Chenildieu", "Cochepaille"}, {"Tholomyes", "Listolier", "Fameuil", "Blacheville", "Favourite", "Dahlia", "Zephine", "Fantine"}, {"Mme. Thenardier", "Thenardier", "Cosette", "Javert", "Boulatruelle", "Eponine", "Anzelma", "Woman2", "Gueulemer", "Babet", "Claquesous", "Montparnasse", "Toussaint", "Brujon"}, {"Fauchelevent", "Mother Innocent", "Gribier"}, {"Pontmercy", "Gillenormand", "Magnon", "Mlle Gillenormand", "Mme. Pontmercy", "Mlle Vaubois", "Lt. Gillenormand", "Marius", "Baroness T"}, {"Jondrette", "Mme. Burgon", "Gavroche", "Mabeuf", "Enjolras", "Combeferre", "Prouvaire", "Feuilly", "Courfeyrac", "Bahorel", "Bossuet", "Joly", "Grantaire", "Mother Plutarch", "Child1", "Child2", "Mme. Hucheloup"}}`

Visualize the detected communities in two different ways:

Plot the adjacency matrix, reordered to show the community structure.

The available properties depend on which algorithm was used for community detection. The following are always present:

`"Properties"`

returns all available properties.`"Algorithm"`

returns the algorithm used for community detection.`"Communities"`

returns the list of communities.`"Elements"`

returns the vertices of the graph.`"ElementCount"`

returns the vertex count of the graph.

These are present for hierarchical clustering methods:

`"HierarchicalClusters"`

returns the clustering in a format compatible with the Hierarchical Clustering standard package.**Note:**Isolated vertices may not be included.`"Merges"`

represents the hierarchical clustering as a sequence of element merges. Elements are represented by their integer indices, and higher indices are introduced for the subclusters formed by the merges. This format is similar to the one used by MATLAB and many other tools.**Note:**Isolated vertices may not be included.`"Tree"`

gives a binary tree representation of the merges.**Note:**Isolated vertices may not be included.

Additionally, the following, and other, algorithm-specific properties may be present:

`"Modularity"`

is a list of modularities for each step of the algorithm, or a single-element list containing the modularity corresponding to the returned clustering. What constitutes a step depends on the particular algorithm.

The `"RemovedEdges"`

property is specific to the `"EdgeBetweenness"`

method, and isn’t present for `"Walktrap"`

.

`{"Valjean" \[UndirectedEdge] "Myriel", "Valjean" \[UndirectedEdge] "Mlle Baptistine", "Valjean" \[UndirectedEdge] "Mme. Magloire", "Gavroche" \[UndirectedEdge] "Valjean", "Gavroche" \[UndirectedEdge] "Javert", "Thenardier" \[UndirectedEdge] "Fantine", "Bamatabois" \[UndirectedEdge] "Javert", "Bossuet" \[UndirectedEdge] "Valjean", "Montparnasse" \[UndirectedEdge] "Valjean", "Gueulemer" \[UndirectedEdge] "Gavroche"}`

Multiple properties may be retrieved at the same time.

Compare the two clusterings:

Visualize the hierarchical clustering using the Hierarchical Clustering Package.

```
DendrogramPlot[cl1["HierarchicalClusters"],
LeafLabels -> (Rotate[#, Pi/2] &), ImageSize -> 750,
AspectRatio -> 1/2]
```

Hierarchical community structures can also be obtained as a vertex-weighted tree graph.

This tree can be supplied as input to `Dendrogram`

.

The graph colouring problem is assigning “colours” or “labels” to the vertices of a graph so that no two adjacent vertices will have the same colour. Similarly, edge colouring assigns colours to edges so that adjacent edges never have the same colour.

IGraph/M represents colours with the integers `1, 2, …`

. Edge directions and self-loops are ignored.

These function will find a colouring of the graph using a fast heuristic algorithm. The colouring may not be minimal. Edge directions are ignored.

Compute a vertex colouring of a Mycielski graph.

`IGVertexColoring`

returns a list of integers, each representing the colour of the vertex that is in the same position in the vertex list.

Associate the colours with vertex names.

Visualize the colours using IGraph/M’s property mapping functionality. See the *Property handling functions* documentation section for more information.

```
Graph[g, VertexSize -> 1/3, EdgeStyle -> Gray] //
IGVertexMap[ColorData[97], VertexStyle -> IGVertexColoring]
```

Visualize an edge colouring of the same graph.

```
Graph[g, GraphStyle -> "ThickEdge", EdgeStyle -> Opacity[0.7],
VertexStyle -> Black] //
IGEdgeMap[ColorData[106], EdgeStyle -> IGEdgeColoring]
```

Compute a checkerboard-like colouring of a three-dimensional grid graph.

```
IGVertexMap[ColorData[97], VertexStyle -> IGVertexColoring,
Graph3D@GridGraph[{4, 4, 4}, VertexSize -> 0.8]]
```

Compute a colouring of a Voronoi mesh.

Compute a colouring of the map of African countries.

```
countries = CountryData["Africa"];
borderingQ[c1_, c2_] := MemberQ[c1["BorderingCountries"], c2]
graph = RelationGraph[borderingQ, countries];
```

```
GeoGraphics@
MapThread[{GeoStyling[{Opacity[0.5], #2}],
Polygon[#1]} &, {countries,
ColorData[97] /@ IGVertexColoring[graph]}]
```

These functions find a colouring with \(k\) or fewer colours. They work by transforming the colouring into a satisfiability problem and using `SatisfiabilityInstances`

.

The available option values are:

forces the given vertices to distinct and increasing colours. Normally, the vertices of a clique are given (which require as many colours as the size of the clique). The main purpose of this option is to reduce the number of redundant solutions of the equivalent SAT problem, and thus improve performance. When using edge colouring functions, a set of edges should be passed.

`"ForcedColoring" -> "MaxDegreeClique"`

attempts to find a clique containing a maximum degree vertex, and forces colours on the clique members. On hard problems it may perform orders of magnitude better than`"ForcedColoring" -> None`

.`"ForcedColoring" -> "LargestClique"`

finds a largest clique, and forces colours on the clique members.`"ForcedColoring" -> None`

does not force any colours. It is usually the fastest choice for easy problems.

The default setting for `"ForcedColoring"`

is `"MaxDegreeClique"`

.

The Moser spindle is not 3-colourable, so no solution is returned.

Find a 4-colouring of the Moser spindle …

… and visualize it.

```
Graph[moser, GraphStyle -> "BasicBlack", VertexSize -> Large] //
IGVertexMap[ColorData[112],
VertexStyle -> (First@IGKVertexColoring[#, 4] &)]
```

Find a 4-edge-colouring of the Petersen graph.

```
PetersenGraph[GraphStyle -> "ThickEdge", EdgeStyle -> Opacity[2/3]] //
IGEdgeMap[ColorData[112],
EdgeStyle -> (First@IGKEdgeColoring[#, 4] &)]
```

The following examples illustrate the use of the `"ForcedColoring"`

option. The 6th order Mycielski graph has chromatic number 6. A 6-colouring is easily found even with `"ForcedColoring" -> None`

.

However, showing that the graph is not 5-colourable takes considerably longer.

Forcing colours in the appropriate way reduces the computation time significantly.

`IGMinimumVertexColoring`

and `IGMinimumEdgeColoring`

find minimum colourings of graphs, i.e. they find a colouring with the fewest possible number of colours. The current implementation tries successively larger \(k\)-colourings until it is successful.

`IGMinimumVertexColoring`

and `IGMinimumEdgeColoring`

can use the same `"ForcedColoring"`

option values as `IGKVertexColoring`

and `IGKEdgeColoring`

.

```
WheelGraph[7, GraphStyle -> "BasicBlack", VertexSize -> Large] //
IGVertexMap[ColorData[97], VertexStyle -> IGMinimumVertexColoring]
```

Find a colouring of a large graph.

`{2, 3, 4, 3, 2, 2, 2, 4, 3, 1, 1, 2, 4, 1, 4, 3, 1, 1, 2, 3, 4, 2, 3, \ 4, 3, 1, 2, 4, 2, 1, 3, 2, 4, 2, 3, 2, 1, 1, 2, 4, 4, 3, 2, 3, 2, 3, \ 2, 2, 3, 4, 4, 1, 3, 4, 1, 1, 4, 1, 3, 4, 4, 1, 3, 4, 1, 1, 3, 4, 2, \ 3, 3, 4, 4, 2, 2, 2, 3, 4, 3, 1, 1, 2, 2, 4, 4, 3, 3, 2, 2, 1, 1, 3, \ 2, 1, 1, 1, 1, 2, 1, 1}`

Implement a multipartite graph layout: vertex colouring is equivalent to partitioning the vertices of the graph into groups such that all connections run between different groups, and never within the same group. The colours can be thought of as the indices of groups. `IGMembershipToPartitions`

can be used to convert from a group-index (i.e. membership) representation to a partition representation.

```
multipartiteLayout[g_?GraphQ, separation : _?NumericQ : 1.5,
opt : OptionsPattern[]] :=
Module[{n, partitions, partitionSizes, vertexCoordinates},
partitions = IGMembershipToPartitions[g]@IGMinimumVertexColoring[g];
partitionSizes = Length /@ partitions;
n = Length[partitions];
vertexCoordinates =
With[{hl = N@Sin[Pi/n],
ir = separation If[n == 2, 1/2, N@Cos[Pi/n]]},
Catenate@Table[
RotationTransform[2 Pi/n k][{#, ir} & /@
Subdivide[-hl, hl, partitionSizes[[k]] - 1]],
{k, 1, n}
]
];
IGReorderVertices[Catenate[partitions], g,
VertexCoordinates -> vertexCoordinates, opt]
];
```

Lay out a bipartite graph.

Lay out multipartite graphs.

```
g = RandomGraph[{40, 160}];
multipartiteLayout[g, GraphStyle -> "BasicBlack",
EdgeStyle -> Opacity[0.2]]
```

Compute a minimum colouring of a triangulation. It can be shown, e.g. based on Brooks’s theorem, that any triangulation of a polygon is 3-colourable.

```
mesh = DelaunayMesh[RandomReal[1, {20, 2}],
MeshCellStyle -> {1 -> Black}];
col = IGMinimumVertexColoring@IGMeshCellAdjacencyGraph[mesh, 2];
SetProperty[{mesh, {2, All}}, MeshCellStyle -> ColorData[97] /@ col]
```

Find a minimum edge colouring of a graph.

```
Graph[
GraphData["SixteenCellGraph"],
GraphStyle -> "ThickEdge", EdgeStyle -> Opacity[2/3]
] //
IGEdgeMap[
ColorData[104],
EdgeStyle -> IGMinimumEdgeColoring
]
```

The chromatic number of a graph is the smallest number of colours needed to colour its vertices. The chromatic index, or edge chromatic number, is the smallest number of colours needed to colour its edges.

Find the chromatic number and chromatic index of a graph.

The implementation of `IGChromaticNumber`

and `IGChromaticIndex`

is effectively the following:

`IGPerfectQ`

tests if a graph is perfect. The clique number and the chromatic number is the same for every induced subgraph of a perfect graph.

The current implementation of `IGPerfectQ`

uses the strong perfect graph theorem: it checks that neither the graph nor its complement have a graph hole of odd length.

The clique number and the chromatic number is the same for every induced subgraph.

`IGVertexColoringQ`

checks whether neighbouring vertices of a graph are assigned different colours.

The colours may be given as a list, with the same ordering as `VertexList[graph]`

.

The colours may also be given as an association from vertices to colours.

Any expression may be used for the colours, not only integers.

`IGRandomWalk[]`

takes a random walk over a directed or undirected graph. If the graph is weighted, the next edge to traverse is selected with probability proportional to its weight.

The available options are:

`EdgeWeight`

can be used to override the existing weights of the graph.`EdgeWeight -> None`

will ignore any existing weights.

Traversing self-loops in different directions is considered as distinct probabilities in an undirected graph. Thus vertices 1 and 3 are visited more often in the below graphs than vertex 2:

```
g = Graph[{1 \[UndirectedEdge] 1, 1 \[UndirectedEdge] 2,
2 \[UndirectedEdge] 3, 3 \[UndirectedEdge] 3},
VertexLabels -> "Name"]
```

This is consistent with their degrees:

Convert the graph to a directed version to traverse self-loops only in one direction.

If the walker gets stuck, a list shorter than `steps`

will be returned. This may happen in a non-connected directed graph, or in a single-vertex graph component.

How much time does a random walker spend on each node of a network?

The exact answer can be computed as the leading eigenvector of the process’s stochastic matrix:

Compare the exact answer with a finite sample:

Random walk on a square grid.

```
grid = IGSquareLattice[{50, 50}];
counts = Counts@IGRandomWalk[grid, 1, 5000];
Graph[grid,
VertexStyle ->
Prepend[
Normal[ColorData["SolarColors"] /@ Normalize[counts, Max]],
Black (* colour of unvisited nodes, i.e. default colour *)
],
EdgeShapeFunction -> None,
Background -> Black
]
```

The fraction of nodes reached after \(n\) steps on a grid and a comparable random regular graph.

Generate random spanning trees using loop erased random walks.

```
randomSpanningTree[graph_?GraphQ] :=
Module[{visited = <||>, i = 2, k = 1,
batchSize = 2 VertexCount[graph], walk},
walk = IGRandomWalk[graph, RandomChoice@VertexList[graph],
batchSize];
visited[walk[[1]]] = True;
While[k < VertexCount[graph],
(* register a traversed edge only when it leads to a yet \
unvisited vertex *)
If[! TrueQ[visited[walk[[i]]]],
Sow[walk[[i - 1]] \[UndirectedEdge] walk[[i]]];
visited[walk[[i]]] = True;
k++
];
i++;
(* if the walk has not yet visited all vertices, keep walking *)
If[i > Length[walk],
walk =
Join[walk, Rest@IGRandomWalk[graph, Last[walk], batchSize]]
];
] // Reap // Last // First
]
```

By taking random spanning trees of spatially embedded planar graphs, we can generate mazes.

Take a sample of a large graph using a random walk. The following graph is too large to easily visualize, but visualizing a random-walk-based sample immediately shows signs of a community structure.

```
g = ExampleData[{"NetworkGraph", "AstrophysicsCollaborations"}];
{VertexCount[g], VertexCount@IGGiantComponent[g]}
```

`IGRandomEdgeWalk`

takes a random walk on a graph and returns the list of traversed edges. If the graph is weighted, the next edge to traverse is selected with probability proportional to its weight.

The available options are:

`EdgeWeight`

can be used to override the existing weights of the graph.`EdgeWeight -> None`

will ignore any existing weights.

Take a random walk on a De Bruijn graph, and retrieve the traversed edges.

`{23 \[DirectedEdge] 15, 15 \[DirectedEdge] 18, 18 \[DirectedEdge] 26, 26 \[DirectedEdge] 23, 23 \[DirectedEdge] 15, 15 \[DirectedEdge] 16, 16 \[DirectedEdge] 21, 21 \[DirectedEdge] 8, 8 \[DirectedEdge] 24, 24 \[DirectedEdge] 16, 16 \[DirectedEdge] 20, 20 \[DirectedEdge] 4, 4 \[DirectedEdge] 11, 11 \[DirectedEdge] 6, 6 \[DirectedEdge] 17, 17 \[DirectedEdge] 22, 22 \[DirectedEdge] 12, 12 \[DirectedEdge] 7, 7 \[DirectedEdge] 20, 20 \[DirectedEdge] 6}`

`IGRandomEdgeIndexWalk`

returns the list of indices of the traversed edges instead. This makes it useful for working with multigraphs, as it allows distinguishing between parallel edges.

As an example application, let us consider the following set of affine transformations:

Let us visualize them by showing an initial (black) triangle and its (red) transformation.

These transformations describe the mutual self-similarity structure of two fractal curves, according to the following directed graph. Each edge of the graph corresponds to a transformation.

Let us compute a random walk on this graph, and iteratively apply transformations to the point `{0, 0}`

according to the traversed edges.

```
walk = IGRandomEdgeIndexWalk[graph, 1, 20000];
pts = Rest@FoldList[#2[#1] &, {0., 0.}, trafos[[walk]]];
```

The resulting list of points will approximate the union of the two fractal curves.

The two curves can be separated by filtering points according to which graph vertex the corresponding directed edge targets. For example, if the point was generated by a transformation corresponding to `1 \[DirectedEdge] 2`

, it will belong to curve `2`

.

The technique described here is taken from “Generating self-affine tiles and their boundaries” by Mark McClure.

`IGSIRProcess`

simulates a stochastic version of the well known SIR model of disease spreading. In this model, each node of the network may be in one of three states: susceptible, infected or recovered, denoted by \(S\), \(I\) and \(R\), respectively. A susceptible node with \(k\) infected neighbours becomes infected with rate \(k \beta\), while an infected node recovers with rate \(\gamma\). At the start of the simulation, a random node is chosen to be infected. The simulation runs until no more infected nodes are left.

When performing a single simulation, `IGSIRProcess`

returns a TimeSeries expression of `{s, i, r}`

values. When multiple runs are requested, the resulting time series are combined into a TemporalData expression.

Perform a single SIR simulation:

Plot the results with a legend:

Plot only the number of infected nodes:

```
(* In Mathematica 12.0 and later,
ts["PathComponent", "I"] can also be used. *)
ListLinePlot[
ts["PathComponent", 2],
AxesLabel -> {"time", "infected"}, PlotStyle -> ColorData[97][2]]
```

Find the number of susceptible, infected and recovered nodes at a specific time point:

The ResamplingMethod of the TimeSeries object is set to 0th order interpolation, therefore the last value is used beyond the last available time point.

Perform 100 simulations simultaneously:

Plot the median number of susceptible, infected and recovered nodes:

```
Show[
ListLinePlot[#, PlotStyle -> GrayLevel[0, 0.1],
PlotRange -> {0, VertexCount[g]}],
Quiet@Plot[Median[#[t]], {t, 0, 4}, PlotStyle -> Red]
] & /@ td["PathComponents"] // GraphicsColumn
```

The sum of the three components, \(S+I+R\), always equals the total number of graph nodes.

In the next example, we compare epidemic spreading on a periodic grid, i.e. a network that only has spatially local connections, with a rewired version of the same network which also includes long range links. We rewire 5% of links while ensuring that the graph stays connected.

```
g1 = IGSquareLattice[{30, 30}, "Periodic" -> True];
g2 = IGTryUntil[IGConnectedQ][IGRewireEdges[g1, 0.05]];
```

Generate 1000 simulations for each network.

Plot the histogram of the total duration of the epidemic.

Plot the fraction of recovered nodes at the end of the epidemic.

```
tmax = Max[r1["LastTimes"], r2["LastTimes"]];
Histogram[
{r1["PathComponent", 3]["SliceData", tmax]/VertexCount[g1],
r2["PathComponent", 3]["SliceData", tmax]/VertexCount[g2]} // Quiet,
{0, 1, 0.02},
ChartLegends -> {"grid", "rewired"}
]
```

A graph is said to be *planar* if it can be drawn in the plane without edge crossings.

A useful concept when working with planar graphs is their combinatorial embedding. A combinatorial embedding of a graph is a counter-clockwise ordering of the incident edges around each vertex. IGraph/M represents combinatorial embeddings as associations from vertices to an ordering of their neighbours. Currently, only embeddings of simple graphs are supported.

Some of the planar graph functionality makes use of the LEMON Graph Library.

`IGPlanarQ[graph]`

checks if a graph is planar using the Boyer–Myrvold algorithm.

`IGPlanarQ[embedding]`

checks if a combinatorial embedding is planar. The following are both embeddings of the \(K_4\) complete graph. However, only the first one is planar.

```
emb1 = <|1 -> {2, 3, 4}, 2 -> {1, 4, 3}, 3 -> {2, 4, 1},
4 -> {3, 2, 1}|>;
emb2 = <|1 -> {2, 4, 3}, 2 -> {4, 3, 1}, 3 -> {1, 2, 4},
4 -> {3, 1, 2}|>;
```

The second embedding generates only 2 faces instead of 4, which can be embedded on a torus, but not in the plane (or on a sphere).

Unlike the built-in `PlanarGraphQ`

, `IGPlanarQ`

considers the null graph to be planar.

A simple graph is *maximal planar* if no new edges can be added to it without breaking planarity. Maximal planar graphs are sometimes called *triangulated* graphs or *triangulations*.

The 3-cycle is maximal planar.

The 4-cycle is not because a chord can be added to it without breaking planarity.

Apollonian graphs are maximal planar.

All faces of a maximal planar graph are triangles.

Therefore the edge count \(E\) and the face count \(F\) of a maximal planar graph on more than 2 vertices satisfy \(2E=3F\). Each edge is incident to two faces and each face is incident to three edges.

`IGOuterplanarQ[graph]`

checks if a graph is outerplanar, i.e. if it can be drawn in the plane without edge crossings and with all vertices being on the outer face.

Outerplanar graphs are also called circular planar. They can be drawn without edge crossings and all vertices on a circle. See the documentation of `IGOuterplanarEmbedding`

for an example.

`IGOuterplanarQ[embedding]`

checks if a combinatorial embedding is outerplanar. Not all planar embeddings of an outerplanar graph are also outerplanar embeddings.

Consider the following outerplanar graph …

… and two of its embeddings:

```
emb1 = <|0 -> {1}, 1 -> {2, 0, 4}, 2 -> {1, 3, 4}, 3 -> {2, 4},
4 -> {3, 1, 2}|>;
emb2 = <|0 -> {1}, 1 -> {0, 2, 4}, 2 -> {1, 3, 4}, 3 -> {2, 4},
4 -> {3, 1, 2}|>;
```

They are both planar, but only the second one is outerplanar.

`IGKuratowskiEdges`

finds a Kuratowski subgraph of a non-planar graph. The subgraph is returned as a set of edges. If the graph is planar, `{}`

is returned.

According to Kuratowski’s theorem, any non-planar graph contains a subgraph homeomorphic to the \(K_5\) complete graph or the \(K_{3,3}\) complete bipartite graph. This is called a Kuratowski subgraph.

Generate a random graph, which is non-planar with high probability.

Compute a set of edges belonging to a Kuratowski subgraph.

`{14 \[UndirectedEdge] 19, 11 \[UndirectedEdge] 19, 9 \[UndirectedEdge] 11, 8 \[UndirectedEdge] 20, 8 \[UndirectedEdge] 10, 7 \[UndirectedEdge] 10, 6 \[UndirectedEdge] 7, 5 \[UndirectedEdge] 6, 4 \[UndirectedEdge] 19, 4 \[UndirectedEdge] 5, 3 \[UndirectedEdge] 14, 3 \[UndirectedEdge] 9, 3 \[UndirectedEdge] 7, 2 \[UndirectedEdge] 20, 2 \[UndirectedEdge] 14, 2 \[UndirectedEdge] 11}`

Highlight the Kuratowski subgraph.

Display the Kuratowski subgraph on its own.

By smoothening the Kuratowski subgraph, we obtain either \(K_5\) or \(K_{3,3}\).

For planar graphs, `{}`

is returned.

`IGFaces`

returns the faces of a planar graph, or the faces corresponding to a specific (not necessarily planar) embedding. The faces are represented by a counter-clockwise ordering of vertices. The current implementation ignores self-loops and multi-edges.

The faces of a planar graph are unique if the graph is 3-vertex-connected. This can be checked using `KVertexConnectedGraphQ`

.

If the graph is not connected and has \(C\) connected components, then \(C-1\) faces will be redundant.

In the above-drawn arrangement, the outer faces of the two triangles are the same face. However, one triangle could have been drawn inside of the other. Then the inner face of one would be the same as the outer face of the other. Thus the choice of faces to be eliminated as redundant is arbitrary, and is left up to the user.

`IGFaces`

can also be used with a non-planar combinatorial embedding. The below embeddings both belong to the 4-vertex complete graph, however, only the first is planar.

```
emb1 = <|1 -> {2, 3, 4}, 2 -> {1, 4, 3}, 3 -> {2, 4, 1},
4 -> {3, 2, 1}|>;
emb2 = <|1 -> {2, 4, 3}, 2 -> {4, 3, 1}, 3 -> {1, 2, 4},
4 -> {3, 1, 2}|>;
```

Determine the genus \(g\) of an embedding belonging to a connected graph based on its face count \(F\), vertex count \(V\), and edge count \(E\), using the formula for the Euler characteristic \(2g-2=\chi =V-E+F\).

`IGDualGraph`

returns a dual graph of a planar graph, or the dual corresponding to a specific embedding. The ordering of the dual graph’s vertices is consistent with the result of `IGFaces`

.

Limitations:

Multi-edges and self-loops are currently ignored.

The result is always a simple graph. No multi-edges or self-loops are generated

The dual of a simple 3-vertex-connected graph is simple and unique, thus such graphs are not affected by the above limitations.

```
TableForm[
Table[{CompleteGraph[k], IGDualGraph@CompleteGraph[k]}, {k, 1, 4}],
TableHeadings -> {None, {"graph", "dual"}}
]
```

Currently, if the input is a graph, it must be planar.

If the input is a combinatorial embedding, it does not need to be planar.

Find the dual of a square lattice graph. The dual graph also includes the outer face as a vertex.

The dual is unique if the graph is 3-vertex-connected. This can be verified using `KVertexConnectedGraphQ`

. In this case, `IGDualGraph@IGDualGraph[g]`

is isomorphic to `g`

.

If the graph is not connected, the dual of each component is effectively computed separately.

`IGEmbeddingQ`

checks if an embedding is valid, and whether it belongs to a graph without self-loops and multi-edges.

This is a valid combinatorial embedding of the graph `1 \[UndirectedEdge] 3 \[UndirectedEdge] 2`

.

The following embeddings do not belong to simple (i.e. loop free and multi-edge free) graphs:

The following embedding is not valid because it does not contain the arc `2 \[DirectedEdge] 1`

but it does contain `1 \[DirectedEdge] 2`

.

`IGPlanarEmbedding`

computes a combinatorial embedding of a planar graph. The current implementation ignores self-loops and multi-edges.

The representation of a combinatorial embedding is also a valid adjacency list, thus it can be easily converted back to an undirected graph using `IGAdjacencyGraph`

.

`IGOuterplanarEmbedding`

returns an outerplanar combinatorial embedding of a graph, if it exists. If the corresponding graph is connected, then one face of such an embedding contains all vertices of the graph.

`IGCoordinatesToEmbedding`

computes a combinatorial embedding, i.e. a cyclic ordering of neighbours around each vertex, based on the given vertex coordinates. By default, the coordinates are taken from the `VertexCoordinates`

property.

The embedding can then be used to compute the faces of the graph …

… or can be converted back to coordinates.

If we start with a non-planar graph layout, the embedding will not be planar either.

`IGEmbeddingToCoordinates`

computes the coordinates of a straight-line planar drawing based on the given combinatorial embedding, using Schnyder’s algorithm.

The embedding must be planar.

`IGLayoutPlanar`

computes a layout of a planar graph without edge crossings using Schnyder’s algorithm. The vertex coordinates will lie on an \((n-2) \times (n-2)\) integer grid, where \(n\) is the number of vertices.

Create a random planar graph and lay it out without edge crossings.

`IGLayoutPlanar`

produces a drawing based on the combinatorial embedding returned `IGPlanarEmbedding`

. A combinatorial embedding is a counter-clockwise ordering of the incident edges around each vertex.

The embedding can also be used to directly compute coordinates for a drawing.

The Tutte embedding can be computed for a 3-vertex-connected planar graph. The faces of such a graph are uniquely defined. This embedding ensures that the coordinates of any vertex not on the outer face are the average of its neighbour’s coordinates, thus it is also called barycentric embedding.

`IGLayoutTutte`

supports weighted graphs, and uses the weights for computing barycentres.

The available options are:

`"OuterFace"`

sets the planar graph face to use as the outer face for the layout. The vertices of the face can be given in any order. Use`IGFaces`

to obtain a list of faces.

By default, a largest face is chosen to be the outer one.

We can specify a different outer face manually.

For some graphs, the best result is achieved when the outer face is not chosen to be a largest one.

`IGLayoutTutte`

requires a 3-vertex-connected planar input.

`IGLayoutTutte`

will take into account edge weights. For a weighted graph, the barycenter of neighbours is computed with a weighting corresponding to the edge weights.

A disadvantage of the Tutte embedding is that the ratio of the shortest and longest edge it creates is often very large. This can be partially remedied by first computing an unweighted Tutte embedding, then setting edge weights based on the obtained edge lengths.

```
IGLayoutTutte@
IGEdgeMap[Apply[EuclideanDistance],
EdgeWeight -> IGEdgeVertexProp[VertexCoordinates], pg]
```

By applying a further power-transformation of the weight, we can fine-tune the layout.

```
Manipulate[
IGLayoutTutte[
IGEdgeMap[(EuclideanDistance @@ #)^power &,
EdgeWeight -> IGEdgeVertexProp[VertexCoordinates], pg],
VertexSize -> 1/2
],
{{power, 1}, 0.5, 3}
]
```

The available options are:

`EdgeWeight`

sets either the explicit edge weights, or the mesh property to be used as edge weights. The default value is`MeshCellMeasure`

. Use`None`

to obtain an unweighted graph.

The following example demonstrates finding a shortest path on a geometric mesh.

```
mesh = DiscretizeRegion[
RegionDifference[Rectangle[{0, 0}, {3, 3}],
Rectangle[{0, 1}, {2, 2}]], MaxCellMeasure -> 0.02]
```

`IGMeshGraph`

preserves the vertex coordinates, and uses edge lengths as edge weights by default.

Find the corners.

```
st = First /@ Through[
{MinimalBy, MaximalBy}[VertexList[g],
Norm@PropertyValue[{g, #}, VertexCoordinates] &]
]
```

Highlight the shortest path.

```
HighlightGraph[g,
PathGraph@FindShortestPath[g, First[st], Last[st]],
Frame -> True, FrameTicks -> True
]
```

Find a Hamiltonian path on a mesh.

```
g = IGMeshGraph@DiscretizeRegion[Disk[], MaxCellMeasure -> 1/40];
HighlightGraph[g, PathGraph@FindHamiltonianPath[g],
GraphHighlightStyle -> "DehighlightHide"]
```

Get a spikey as a graph.

The available options for `IGMeshCellAdjacencyGraph`

are:

`VertexCoordinates -> Automatic`

will use the mesh cell centroids as vertex coordinates if the mesh is 2 or 3-dimensional. The default is`VertexCoordinates -> None`

, which does not compute any coordinates.

Compute the connectivity of mesh vertices (zero-dimensional cells).

Compute the connectivity of faces (two-dimensional cells).

Create the graph of a Goldberg polyhedron.

```
IGMeshCellAdjacencyGraph[
BoundaryDiscretizeRegion[Ball[], PrecisionGoal -> 1,
MaxCellMeasure -> 0.5], 2,
VertexCoordinates -> Automatic]
```

Compute the connectivity of faces and edges, and colour nodes based on whether they represent a face or an edge.

```
g = IGMeshCellAdjacencyGraph[
mesh, 2, 1,
VertexSize -> 0.9,
VertexStyle -> {EdgeForm[], {1, _} -> Red, {2, _} -> Black},
EdgeStyle -> Gray]
```

This is a bipartite graph.

The vertex names are the same as the mesh cell indices (see `MeshCellIndex`

).

Colour the faces of the mesh.

```
SetProperty[{mesh, {2, All}},
MeshCellStyle ->
ColorData[100] /@
IGVertexColoring@IGMeshCellAdjacencyGraph[mesh, 2]
]
```

The edge-edge connectivity is identical to the line graph of the vertex-vertex connectivity.

Compute the adjacency matrix of the vertex-vertex connectivity.

Compute the adjacency matrix of the edge-face connectivity.

This is the (non-square) incidence matrix of a bipartite graph. The graph can be reconstructed using `IGBipartiteIncidenceGraph`

.

Paint a Hamiltonian path on triangulation using a gradient of colours.

```
mesh = DiscretizeRegion[Disk[], MaxCellMeasure -> 1/50,
MeshCellStyle -> {1 -> None}];
path = FindHamiltonianPath@IGMeshCellAdjacencyGraph[mesh, 2];
```

```
MeshRegion[
mesh,
MeshCellStyle ->
MapIndexed[#1 -> ColorData["Pastel"][First[#2]/Length[path]] &, path]
]
```

`IGLatticeMesh`

can generate meshes of various periodic tilings. `IGMeshGraph`

and `IGMeshCellAdjacencyGraph`

can be used to convert these to graphs. The primary use case is the easy generation of various lattice graphs.

`IGLatticeMesh[]`

returns the list of available lattices. Let us explore them using a graphical interface.

IGraph/M knows about a subset of the tilings available in `EntityClass["PeriodicTiling", All]`

. Use these entities to obtain additional geometric information about the tilings.

Generate a *kagome lattice* consisting of 6 by 4 unit cells.

Create a hexagonal graph of 4 by 3 cells. Notice that the nodes are labelled with consecutive integers along the translation vectors of the lattice.

```
IGMeshGraph[
IGLatticeMesh["Hexagonal", {4, 3}],
VertexShapeFunction -> "Name", PerformanceGoal -> "Quality"]
```

This specific node labelling allows for the creation of convenient directed lattices.

Create a hexagonal mesh from points that fall within a rectangular region.

Create a hexagonal mesh from points that fall within a hexagonal region.

Create a triangular grid graph in the shape of a hexagon, as the face-adjacency graph of the above mesh.

Create a face adjacency graph of the Cairo pentagonal tiling, and display it along with its mesh.

```
Show[
mesh,
IGMeshCellAdjacencyGraph[mesh, 2,
VertexCoordinates -> Automatic, GraphStyle -> "BasicBlack"],
ImageSize -> Medium
]
```

Compute a colouring of a periodic tiling so that neighbouring cells have different colours.

```
colorMesh[mesh_] :=
SetProperty[{MeshRegion[mesh, MeshCellStyle -> {1 -> White}], {2,
All}},
MeshCellStyle ->
ColorData[8] /@
IGMinimumVertexColoring@IGMeshCellAdjacencyGraph[mesh, 2]
]
```

Explore the face-adjacency graphs of lattices. These correspond to the dual lattice.

```
Manipulate[
IGMeshCellAdjacencyGraph[IGLatticeMesh[type], 2,
VertexCoordinates -> Automatic],
{type, IGLatticeMesh[]}
]
```

Make a maze through the faces of a lattice. We start by finding a spanning tree of the face-edge incidence graph of the lattice.

```
mesh = IGLatticeMesh["Square", Disk[{0, 0}, 9.5],
MeshCellStyle -> {1 | 2 -> GrayLevel[0.9]}];
t = IGRandomSpanningTree@IGMeshCellAdjacencyGraph[mesh, 2, 1];
```

The walls of the maze will be the leaves of this tree which are edges.

We will remove two outer walls to serve as the

Draw the maze.

```
MeshRegion[mesh,
MeshCellStyle ->
Thread[Complement[walls, exits] ->
Directive[AbsoluteThickness[4], Black]],
Epilog -> {Text["\!\(\*
StyleBox[\"\[LongRightArrow]\",\nFontWeight->\"Bold\",\n\
FontColor->RGBColor[1, 0, 0]]\)", #] & /@
PropertyValue[{mesh, exits}, MeshCellCentroid]}
]
```

Create a Moiré pattern by superimposing two rotated hexagonal lattices.

```
m = IGLatticeMesh["Hexagonal", Polygon@CirclePoints[12., 6]];
Manipulate[
Show@Table[
MeshRegion[
TransformedRegion[m, RotationTransform[angle]],
MeshCellStyle -> {2 -> None, 1 -> AbsoluteThickness[1.5]},
PlotRange -> 13 {{-1, 1}, {-1, 1}}
],
{angle, {0, α}}
],
{{α, 0.15}, 0, 0.3}
]
```

Proximity graphs are connectivity structures of geometric points based on geometric criteria. IGraph/M implements several proximity graphs for points in two-dimensional Euclidean space.

`IGDelaunayGraph[points]`

creates computes the Delaunay graph of the given points in one, two or three dimensions. It is equivalent to `IGMeshGraph@DelaunayMesh[points]`

, but it is faster and it supports collinear points in 2D and coplanar points in 3D.

`IGDelaunayGraph`

works in 1D, 2D and 3D.

IGDelaunayGraph works with collinear points in 2D …

… or coplanar points in 3D.

IGDelaunayGraph takes all the usual graph options.

When there is more than one valid Delaunay triangulation, only one is returned.

Find and plot an Euclidean minimum spanning tree of a set of points in three dimensions.

```
dg = IGDelaunayGraph[pts];
IGTakeSubgraph[dg,
IGSpanningTree@
IGEdgeMap[Apply[EuclideanDistance],
EdgeWeight -> IGEdgeVertexProp[VertexCoordinates], dg]
]
```

The lune-based β skeleton connects two points \(A\) and \(B\) when the intersection of two disks (a lune) having \(A\) and \(B\) on its boundary contains no other points.

For \(\beta \leq 1\), the lune is defined by disks of radius \(\text{\textit{AB}}/(2\beta )\).

For \(\beta \geq 1\), the lune is defined by disks of radius \(\beta \text{\textit{AB}}/2\).

For \(\beta \geq 1\), the definition generalizes to higher dimensions too. `IGLuneBetaSkeleton`

supports 2D and 3D point sets.

For \(\beta \geq 1\), the β skeleton is a subgraph of the Delaunay graph, thus its edges do not cross. For \(\beta \leq 2\), it contains the Euclidean minimum spanning tree, thus it is connected. For \(\beta >2\), it is typically disconnected.

The implementation of β skeleton computation is efficient only for \(\beta \geq 1\).

β skeletons can be used to reconstruct a shape from a set of points.