Definition:Adjacency Matrix

Definition

An adjacency matrix is a matrix which describes a graph by representing which vertices are adjacent to which other vertices.

If $G$ is a graph of order $n$, then its adjacency matrix is an $n \times n$ square matrix, where each row and column corresponds to a vertex of $G$.

The element $a_{i j}$ of such a matrix specifies the number of edges from vertex $i$ to vertex $j$.

An adjacency matrix for a simple graph and a loop-digraph is a logical matrix, i.e. one whose elements are all either $0$ or $1$.

An adjacency matrix for an undirected graph is symmetrical about the main diagonal.

This is because if vertex $i$ is adjacent to vertex $j$, then $j$ is adjacent to $i$.

An adjacency matrix for a weighted graph or network contains the weights of the edges.

Examples

Simple Graph

The elements of an adjacency matrix of a simple graph are $0$ and $1$, and the diagonal elements are all zero:

$\qquad \begin{pmatrix} 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix}$

Multigraph

The elements of an adjacency matrix of a multigraph are integers, and the diagonal elements are all zero:

$\qquad \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 & 3 & 0 \\ \end{pmatrix}$

Digraph

An adjacency matrix for a directed graph is no longer symmetrical about the main diagonal:

$\qquad \begin{pmatrix} 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ \end{pmatrix}$

Loop-Graph

The elements on the main diagonal of an adjacency matrix for a loop-graph are not all non-zero:

$\qquad \begin{pmatrix} 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 2 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix}$

Note that some other treatments of this subject require that a loop contributes a value of $1$ to the vertex to which it is incident.

Loop-Multigraph

The elements of the adjacency matrix of a loop-multigraph are integers, and this time some of the diagonal elements are non-zero:

$\qquad \begin{pmatrix} 0 & 2 & 0 & 0 & 0 & 0 \\ 2 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 \\ 0 & 1 & 1 & 2 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 1 & 3 & 0 \\ \end{pmatrix}$

Loop-Digraph

An adjacency matrix for a loop-digraph is not only not symmetrical about the main diagonal, it also has entries on that main diagonal:

$\qquad \begin{pmatrix} 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ \end{pmatrix}$

Network

An adjacency matrix for a general network can have any numbers in any of its elements:

$\qquad \begin{pmatrix} 0 & 1.2 & 0 & 0.5 \\ 0 & 0 & 0 & 3.4 \\ 0 & 0 & 0 & 0 \\ 4.1 & 0 & 10.1 & 0 \\ \end{pmatrix}$