A matrix representation of graphs and its spectrum as a graph invariant

Details A matrix representation of graphs and its spectrum as a graph invariant A matrix representation of graphs and its spectrum as a graph invariant

2005-05-05

arxiv.org/abs/quant-ph/0505026v2

The Electronic Journal of Combinatorics 13 (2006), #R34

Physics

Quantum Physics

14 pages

Scientific paper

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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