Suppose we have two (finite, simple, undirected) graphs, what can we say about these graphs if they have similar adjacency matrices?
Observations to begin with:
If $G_1$ and $G_2$ are isomorphic, then they have similar adjacency matrices, $A_1$ and $A_2$. In fact, they are similar in an even stronger sense: they satisfy $A_1=PA_2P^{-1}$, where $P$ is a permutation matrix.
The following non-isomorphic graphs, have similar adjacency matrices:
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Similarity of adjacency matrices is an equivalence relation on the set of $n$-vertex graphs.
Graphs with similar adjacency matrices must be isospectral graphs.