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

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.Comment: 5 pages, 3 figure

Nonexistence of Certain Skew-symmetric Amorphous Association Schemes

An association scheme is amorphous if it has as many fusion schemes as possible. Symmetric amorphous schemes were classified by A. V. Ivanov [A. V. Ivanov, Amorphous cellular rings II, in Investigations in algebraic theory of combinatorial objects, pages 39--49. VNIISI, Moscow, Institute for System Studies, 1985] and commutative amorphous schemes were classified by T. Ito, A. Munemasa and M. Yamada [T. Ito, A. Munemasa and M. Yamada, Amorphous association schemes over the Galois rings of characteristic 4, European J. Combin., 12(1991), 513--526]. A scheme is called skew-symmetric if the diagonal relation is the only symmetric relation. We prove the nonexistence of skew-symmetric amorphous schemes with at least 4 classes. We also prove that non-symmetric amorphous schemes are commutative.Comment: 10 page