What can we say about two graphs if they have similar adjacency matrices?

Asked Sep 01 '13 at 00:42

Active Apr 04 '20 at 08:06

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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:

$\qquad\qquad\qquad$
Similarity of adjacency matrices is an equivalence relation on the set of $n$-vertex graphs.
Graphs with similar adjacency matrices must be isospectral graphs.

edited Jun 12 '20 at 10:38

Community

asked Sep 01 '13 at 00:42

Rebecca J. Stones

2

Given two such graphs you can say that they have the same number of edges and triangles. – Jernej Sep 01 '13 at 22:22
1

Also if your graphs are regular then they have the same number of spanning trees as well. – Jernej Sep 01 '13 at 22:44

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