Consider a real symmetric matrix. Such a matrix can be considered as an adjacency matrix of a graph, and in fact may be identified with the graph itself.
Now consider the equivalence class of the relation of isomorphism between graphs. All the graph invariants are same for all the graphs in the same equivalence class. Similarly all the algebraic properties for the adjacency matrices are same, i.e. rank, determinant, spectrum etc. This is just because any two graphs are isomorphic translates to saying that the adjacency matrices are similar(Or even more strongly it translates to that they are permutation equivalent matrices. Or is there some even more stronger term, because they are related as $B=PAP^T$ and not simply as $B=PAQ$ for permutation matrices $P,Q$?)
However I feel, that there is some more deeper interpretation here in the connection between adjacency matrices and the associated graph. All isomorphic graphs correspond to the same operator and can a deeper interpretation be derived from that? Specifically why do the graph theoretic properties translate to algebraic properties?