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{\bf David Emms, Edwin R. Hancock, Simone Severini and Richard C. Wilson}
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{\bf A Matrix Representation of Graphs and its Spectrum as a Graph Invariant}
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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.
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