Checking for Graph Isomorphism

Once you found that two graphs are isomorphic, you can construct the adjacency matrix for both graphs. That is, once you know which nodes in $G_1$ and in $G_2$ match, the adjacency matrix for $G_1$ will be identical to the one for $G_2$ (using the same matching order of vertices.)

For example, the adjacency matrices for graphs $G_1$ and $G_2$ from slide 48 are respectively:

$\text{adj}(G_1) = \begin{bmatrix} & \boldsymbol 1 & \boldsymbol 2 & \boldsymbol 3 & \boldsymbol 4 \\ \boldsymbol 1 & 0 & 1 & 1 & 1 \\ \boldsymbol 2 & 1 & 0 & 0 & 0 \\ \boldsymbol 3 & 1 & 0 & 0 & 1 \\ \boldsymbol 4 & 1 & 0 & 1 & 0 \end{bmatrix}$

and

$\text{adj}(G_2) = \begin{bmatrix} & \boldsymbol A & \boldsymbol C & \boldsymbol B & \boldsymbol D \\ \boldsymbol A & 0 & 1 & 1 & 1 \\ \boldsymbol C & 1 & 0 & 0 & 0 \\ \boldsymbol B & 1 & 0 & 0 & 1 \\ \boldsymbol D & 1 & 0 & 1 & 0 \end{bmatrix}$

These notes by Miriam Briskman are licensed under CC BY-NC 4.0 and based on sources.