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A short note on a short remark of Graham and Lovász

Let D be the distance matrix of a connected graph G and let nn(G), np(G) be the number of strictly negative and positive eigenvalues of D respectively. It was remarked in [1] that it is not known whether there is a graph for which np(G) > nn (G). In this note we show that there exists an infinite number of graphs satisfying the stated inequality, namely the conference graphs of order> 9. A large representative of this class being the Paley graphs.The result is obtained by derving the eigenvalues of the distance matrix of a strongly-regular graph.

preprint2013arXivOpen access
Jernej Azarija
math.CO
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Jernej Azarija

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