Editing Adjacency matrix (section)

===Isomorphism and invariants===
Suppose two directed or undirected graphs {{math|''G''<sub>1</sub>}} and {{math|''G''<sub>2</sub>}} with adjacency matrices {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}} are given. {{math|''G''<sub>1</sub>}} and {{math|''G''<sub>2</sub>}} are [[graph isomorphism|isomorphic]] if and only if there exists a [[permutation matrix]] {{mvar|P}} such that
: <math>P A_1 P^{-1} = A_2.</math>

In particular, {{math|''A''<sub>1</sub>}} and {{math|''A''<sub>2</sub>}} are [[Similar (linear algebra)|similar]] and therefore have the same [[Minimal polynomial (linear algebra)|minimal polynomial]], [[characteristic polynomial]], [[eigenvalues]], [[determinant]] and [[Trace (matrix)|trace]]. These can therefore serve as isomorphism invariants of graphs. However, two graphs may possess the same set of eigenvalues but not be isomorphic.<ref>[[Godsil, Chris]]; [[Gordon Royle|Royle, Gordon]] ''Algebraic Graph Theory'', Springer (2001), {{ISBN|0-387-95241-1}}, p.164</ref> Such [[linear operator]]s are said to be [[isospectral]].