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Isomorphism and invariants

Suppose two directed or undirected graphs G and G with adjacency matrices A and A are given. G and G are isomorphic if and only if there exists a permutation matrix P such that

In particular, A and A are similar and therefore have the same minimal polynomial, characteristic polynomial, eigenvalues, determinant and trace. These can therefore serve as isomorphism invariants of graphs. However, two graphs may possess the same set of eigenvalues but not be isomorphic. Such linear operators are said to be isospectral.