A short note on a short remark of Graham and Lovلsz

Let D be the distance matrix of a connected graph G and let n − ( G ) , n + ( G ) be the numbers of strictly negative and positive eigenvalues of D respectively. It was remarked in Graham and Lovász (1978) that it is not known whether there is a graph for which n + ( G ) > n − ( G ) . In this note we show that there exist an infinite number of graphs satisfying the stated inequality, namely the conference graphs of order >9, a major representative of this class being the Paley graphs. The result is obtained by deriving the eigenvalues of the distance matrix of a strongly regular graph.