Non isomorphic graph and spectrum of a adjacency matrix

Question

Are two non isomorphic graphs with the same spectrum of adjacency matrix possible?

Answer 1

The complete bipartite graph $K_{1,4}$ and the disjoint union of the cycle $C_4$ are the smallest pair of examples.

Answer 2

It is very likely possible. They called that kind of graph "isospectral" or "cospectral".

Check the Table1 in the following paper.

https://arxiv.org/pdf/1008.3646.pdf

It says there are total 274668 non-isomorphic graphs that have 9 vertices. But 51039 of them have the same eigenvalues according to Adjacency. Btw, if you use normalized laplacian instead of Adjacency, there are fewer (just 1092 pairs) cospectral graphs.