A short note on a short remark of Graham and Lov\'{a}sz

TL;DR

This paper demonstrates that there are infinitely many graphs, specifically conference and Paley graphs, where the number of positive eigenvalues of their distance matrix exceeds the number of negative eigenvalues, addressing a previously open question.

Contribution

It proves the existence of infinitely many graphs with more positive than negative eigenvalues in their distance matrix, using eigenvalue analysis of strongly-regular graphs.

Findings

Conference graphs of order > 9 have np(G) > nn(G).

Paley graphs are a large class of such graphs.

Eigenvalues of the distance matrix are derived for strongly-regular graphs.

Abstract

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.