# nonisomorphic cospectral regular graphs

Is there any non-isomorphic cospectral regular graphs? Please suggest some examples or reference. Cospectral graphs are those graph which possess same set of spectrum.

• I expect the answer in negative. but do not have justification for it. – G_0_pi_i_e Mar 1 '16 at 6:15

Yes, there are. Take an $n\times n$ Latin square $L$ and construct from it a graph as follows. The vertices of the graph are the $n^2$ positions in the Latin square. Two positions are adjacent in our graph if they are in the same row, the same column, or have the same entry. This gives a regular graph with valency $3n-3$. The eigenvalues of the graph are $3n-3$, $n-3$ and $-3$ with respective multiplicities $1$, $3n-3$ and $n^2-3n+2$. To get your first examples, take the two Latin squares arising as the multiplication tables of the groups of order four.