Rational orthogonal matrices and isomorphism of graphs

Let On(Q) and Sn(Z) be the set of all n by n orthogonal matrices with rational entries and the set of all n by n symmetric matrices with integer entries, respectively. We consider the following question: Given A∈Sn(Z). When does QTAQ∈Sn(Z) with Q∈On(Q) imply that Q is a signed permutation matrix? Wang and Yu [15] show that the answer to the above question is yes whenever the discriminant Δ(A) of the characteristic polynomial of A is odd and square-free. However, the above result cannot be applied for the adjacency matrix of a simple graph, since Δ(A) can never be odd in this situation. We give a new condition under which the answer is still yes for the adjacency matrices of graphs. The result was achieved by employing some tools from our recent work on the generalized spectral characterizations of graphs, which is somewhat unexpected. Some applications of the above result are also provided.