A note on non-R-cospectral graphs

Two graphs G and H are called R-cospectral if A(G)+yJ and A(H)+yJ where A(G), A(H) are the adjacency matrices of G and H, respectively, J is the all-one matrix) have the same spectrum for all y is an element of R. In this note, we give a necessary condition for having R-cospectral graphs. Further, we provide a sufficient condition ensuring only irrational orthogonal similarity between certain cospectral graphs. Some concrete examples are also supplied to exemplify the main results.