# Adjacency matrix ¶

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This may also be called the boolean matrix (rarely) or Matrice d’adjacence/Matrice booléenne. This is a matrix vertex by vertex, and the values are either 0 or 1. If we are at row=A, col=B

• 1: $A \to B$ exists
• 0: $A \to B$ do not exist

Notes

• in an undirected graph, the matrix is symmetric.
• if the vertex is looping, we are adding a 2 on the diagonal (and that's can't be called boolean matrix anymore)

This matrix is quite convenient because if $A$ is the adjacency matrix, then $A^p$ is the adjacent matrix after $p$ transition. You can use that to know, after $p$ steps, where you can go.

## Example 1 ¶

The adjacency matrix for

is

$\displaylines{ \hspace{0.7cm}\begin{array}{} a&b&c&d&h&i \end{array} \ \ \ \\ \begin{array}{} a\\b\\c\\d\\h\\i \end{array} \begin{pmatrix} 0 & 1 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix} }$

## Example 2 ¶

The adjacency matrix for

is

$\displaylines{ \hspace{0.7cm}\begin{array}{} a&b&c&d&h&i \end{array} \ \ \ \\ \begin{array}{} a\\b\\c\\d\\h\\i \end{array} \begin{pmatrix} 0 & 1 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ \end{pmatrix} }$

You may have noticed, but if we add the edge (d,i), we got a Wheel ($W_{6}$).