Determine whether the two graphs $G$ and $G'$ are isomorphic given their adjacency matrices:
$$G=\left[\begin{array}{llllllll} 0 & 1 & 1 & 0 & 0 & 0 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 1 \\ 1 & 1 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 & 1 & 1 & 0 \end{array}\right]$$
and
$$G' = \left[\begin{array}{llllllll} 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \end{array}\right]$$
We have learned several methods to solve a question like this in class, for example by creating sets of the relations and defining an onto and one-to-one function. Or visually by drawing the graphs. This works great for what we have had so far, but i have never had to do it with two such large matrices. This is a question from a previous exam and I was wondering whether there was a faster and more effective way to view if they are isomorphic or not since I would like to not have to use such a long time if a question like this comes on my exam.
I don't think there is an easy way of doing it since graph ismorphism is a hard problem (it's not know which class it's in) .
however there are a couple of heuristics to help, counting degrees, matching by degrees, counting loops, or finding any kind of pattern
in your example in the second matrix it's clear that the graph is bipartite with one side having being the odd indexed nodes and the other the even indexed nodes (notice that there is only two different rows that keep repeating), however the first graph is clearly not bipartite so the answer is easy.