I just took an exam.
One of the open ended questions was to prove that two graphs are isomorphic. I set about proving that the mapping $$\phi : G \rightarrow H $$ is bijective. Let's assume I did that part of the question correctly.
Then, in order to show that adjacency and non-adjacency were preserved, I wrote out the adjacency matrix of each graph, and showed they were equal.
Is writing out the adjacency matrix of each graph and showing that they are equal enough to show that adjacency and non-adjacency are preserved? I am not concerned about how mathematically rigorous this is or isn't, considering most people in my class simply said "a cycle exists, and the vertices and edges are shared, so it is isomorphic."
I am concerned if it is mathematically sufficient to show adjacency and non-adjacency by showing that the adjacency matrices are the same. After all, with an adjacency matrix, we can see what is and is not adjacent, so is this a valid step in proving the graphs are isomorphic?