Isospectral Graphs
Two graphs are called isospectral or cospectral if the adjacency matrices of the graphs have equal multisets of eigenvalues.
Isospectral graphs need not be isomorphic, but isomorphic graphs are always isospectral. The smallest pair of nonisomorphic cospectral undirected graphs is {K1,4, C4 U K1}, comprising the 5-vertex star and the graph union of the 4-vertex cycle and the single-vertex graph, as reported by Collatz and Sinogowitz in 1957. The smallest pair of nonisomorphic cospectral polyhedral graphs are enneahedra with eight vertices each.
Almost all trees are cospectral, i.e., the share of cospectral trees on n vertices tends to 1 as n grows.
Read more about this topic: Spectral Graph Theory