Two graphs $G_1$ and $G_2$ that are cospectral (eigenvalue multisets from their adjacency matrices are the same), do not have to be isomorphic. The pair of cospectral graphs that serve as the smallest counterexample to isomorphism are the disjoint graph union of $C_4 \cup K_1$ and the star graph $S_5$.
What is the smallest counterexample known when we require that $G_1$ and $G_2$ are connected?