Counterexample for graph isomorphism using eigenvalue multiplicity (connected graphs)

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?

Solution 1:

Harary, King, Mowshowitz, Read, Cospectral graphs and digraphs, Bull London Math Soc 3 (1971) 321-328, give an example of two connected, cospectral, nonisomorphic graphs on 6 vertices, and claim, on the basis of exhaustive search, that it's the smallest.

EDIT: Here's the example. Let the vertices be $A,B,C,D,E,F$. For one graph, join $A$ to each of the other vertices, then join $BC$ and $DE$. For the other, draw a path $ABCDE$, then join $F$ to $B,C,D$.