  We use the line digraph construction to associate an orthogonal matrix with
each graph. From this orthogonal matrix, we derive two further matrices. The
spectrum of each of these three matrices is considered as a graph invariant.
For the first two cases, we compute the spectrum explicitly and show that it is
determined by the spectrum of the adjacency matrix of the original graph. We
then show by computation that the isomorphism classes of many known families of
strongly regular graphs (up to 64 vertices) are characterized by the spectrum
of this matrix. We conjecture that this is always the case for strongly regular
graphs and we show that the conjecture is not valid for general graphs. We
verify that the smallest regular graphs which are not distinguished with our
method are on 14 vertices.
