Definition

Let $G=(V,E)$ be a graph with vertex set $V=\lbrace v_1,\ldots, v_n \rbrace$ and edge set $E$ . The adjacency matrix $M_G=(m_{ij})$ of $G$ is defined as follows: $M_G$ is an $n\times n$ matrix such that

For example, the adjacency matrix of the following graph

is

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

Remarks. Let $G$ be a graph and $M_G$ be its adjacency matrix.

• $M_G$ is symmetric with $0$ 's in its main diagonal.
• The sum of the cells in any given column (or row) is the degree of the corresponding vertex. Therefore, the sum of all the cells in $M_G$ is twice the number of edges in $G$ .
• $M_G=\mathbf{1}-I$ iff $G$ is a complete graph. Here, $\mathbf{1}$ is the matrix whose entries are all $1$ and $I$ is the identity matrix.
• If we are given a symmetric matrix $M$ of order $n$ whose entries are either $1$ or $0$ and whose entries in the main diagonal are all $0$ , then we can construct a graph $G$ such that $M=M_G$ .

Generalizations

The above definition of an adjacency matrix can be extended to multigraphs (multiple edges between pairs of vertices allowed), pseudographs (loops allowed), and even directed pseudographs (edges are directional). There are two cases in which we can generalize the definition, depending on whether edges are directional.

1. (Edges are not directional).

   Since a multigraph is just a special case of a pseudograph, we will define $M_G$ for a pseudograph $G$ . Let $G=(V,E)$ be a pseudograph with $V=\lbrace v_1,\ldots,v_n\rbrace$ The adjacency matrix $M_G=(m_{ij})$ of $G$ is an $n\times n$ matrix such that $m_{ij}$ is the number of edges whose endpoints are $v_i$ and $v_j$ . Again, $M_G$ is symmetric, but the main diagonal may contain non-zero entries, in case there are loops.

2. (Edges are directional).

   Since a digraph is a special case of a directed pseudograph, we again define $M_G$ in the most general setting. Let $G=(V,E)$ be a directed pseudograph with $V=\lbrace v_1,\ldots,v_n\rbrace$ and $E\subseteq V\times V\times (\mathbb{N}\cup \lbrace 0\rbrace)$ . The adjacency matrix $M_G$ of $G$ is an $n\times n$ matrix such that $$m_{ij}=|\lbrace k\mid (v_i,v_j,k)\in E\rbrace|.$$ In other words, $m_{ij}$ is the number of directed edges from $v_i$ to $v_j$ .

Remarks

• If $G$ is a multigraph, then the entries in the main diagonal of $M_G$ must be all $0$ .
• If $G$ is a graph, then $M_G$ corresponds to the original definition given in the previous section.
• If $G$ is a digraph, then entries $M_G$ consists of $0$ 's and $1$ 's and its main diagonal consists of all $0$ 's.
• Given any square matrix $M$ , there is a directed pseudograph $G$ with $M=M_G$ . In addition, $M$ corresponds to adjacency matrix of various types of graphs if appropriate conditions are imposed on $M$
• Generally, one can derive a pseudograph from a directed pseudograph by ``forgetting'' the order in the ordered pairs of vertices. If $G$ is a directed pseudograph and $G'$ is the corresponding derived pseudograph. Let $M_G=(m_{ij})$ and $M_{G'}=(n_{ij})$ , then $n_{ij}=m_{ij}+m_{ji}$ .

  In the language of category theory, the above operation is done via a forgetful functor (from the category of directed pseudographs to the category of pseudographs). Other forgetful functors between categories of various types of graphs are possible. In each case, the forgetful functor has an associated operation on the adjacency matrices of the graphs involved.