adjacency matrix

Let $G$ be an undirected graph with $n$ vertices. The adjacency matrix of $G$ is the matrix $A = (a_{i j})$ with
$a_{i j} =$ number of edges between vertex $i$ and $j$

If $G$ is directed graph, $a_{i j}$ is the number of directed edges $i \to j$ .

If $G$ is a weighted graph, $a_{i j}$ is the sum over all edges outgoing from $i$ the weights of those edges.

Properties

symmetric matrix if $G$ is undirected

Example

Let $G$ be the following graph:
Pasted image 20260409095759.png|200
Its adjacency matrix as an UNDIRECTED graph is

A = [\begin{matrix} 0 & 1 & 1 & 2 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 2 & 1 & 1 & 0 \end{matrix}]

Whereas as a directed graph, it is

A = [\begin{matrix} 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{matrix}]