BUILT-IN MATHEMATICA SYMBOL

AdjacencyMatrix

AdjacencyMatrix[g]
gives the vertex-vertex adjacency matrix of the graph g.

MORE INFORMATION

AdjacencyMatrix returns a SparseArray object, which can be converted to an ordinary matrix using Normal.

An entry of the adjacency matrix is the number of directed edges from vertex to vertex .

The diagonal entries count the number of loops for vertex .

An undirected edge is interpreted as two directed edges with opposite directions.

The vertices are assumed to be in the order given by VertexList[g].

The adjacency matrix for a graph will have dimensions , where is the number of vertices.

EXAMPLES

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Basic Examples (2)

The adjacency matrix of an undirected graph:

The adjacency matrix of a directed graph:

The adjacency matrix of an undirected graph:

In[1]:=

Out[1]=

In[2]:=

Out[2]//MatrixForm=

The adjacency matrix of a directed graph:

In[1]:=

Out[1]=

In[2]:=

Out[2]//MatrixForm=

Scope (4)

The adjacency matrix of an undirected graph is symmetric:

The adjacency matrix of a directed graph can be unsymmetric:

The adjacency matrix of the graph with self-loops has diagonal entries:

AdjacencyMatrix works with large graphs:

Use MatrixPlot to visualize the matrix:

Applications (7)

Compute the degree for an undirected graph from its adjacency matrix:

Compute the in-degree for a directed graph from its adjacency matrix:

Compute the out-degree for a directed graph from its adjacency matrix:

Count the number of paths between all vertices in at most

steps for a directed graph:

There are two paths from 1 to 5 in two steps:

Count the number of paths from

to

in at most

steps for a directed graph:

Compute the cocitation matrix, where the cocitation for two vertices is the number of common ancestors:

The cocitation between

and

:

Compute the coupling matrix, where the coupling between two vertices is the number of common descendants:

The coupling between

and

:

Properties & Relations (13)

Rows and columns of the adjacency matrix follow the order given by VertexList:

Use VertexIndex to find the matrix row and column corresponding to a pair of vertices:

Check whether

and

are adjacent vertices:

Compare with EdgeQ:

An undirected graph has a symmetric adjacency matrix:

Use AdjacencyGraph to construct a graph from an adjacency matrix:

The main diagonal of the adjacency matrix for any graph without loops has all zero entries:

The number of rows or columns of the adjacency matrix is equal to the number of vertices:

A d-regular graph g is connected iff the multiplicity of its d eigenvalue is one:

The graph is 3-regular:

The multiplicity

is one so it is connected:

For a complete graph all entries outside the diagonal are ones in the adjacency matrix:

A complete

-partite graph has zero diagonal block entries:

A TuranGraph is bipartite:

A StarGraph has ones in the first column and first row only:

For a path, rows of an adjacency matrix will contain one or two elements:

The adjacency matrix of a line graph can be computed by its IncidenceMatrix:

Possible Issues (1)

An empty graph has no adjacency matrix: