Addressing the Petersen graph

In a 1971 paper motivated by a problem on message routing in a communications network, Graham and Pollack propose a scheme for addressing the vertices of a graph G by N-tuples of three symbols in such a way that distances between vertices may readily be determined from their addresses. They observe that N h(D), the maximum of the number of positive and the number of negative eigenvalues of the distance matrix D of G. A result from a paper of Gregory, Shader and Watts (1999) yields a necessary condition for equality to occur. As an illustration, it is shown here that N ? h(D) = 5 for all addressings of the Petersen graph. Also, an optimal addressing of the Petersen graph by 6-tuples is given. MSC: 05C50, 05C20 Keywords: Eigenvalues; Addressing; Distance matrix Throughout the paper, G will always denote a finite, connected, simple graph with vertices 1; 2; : : : ; n and n \Theta n adjacency matrix A. Terminology not defined here may be found in the book by van Lint and Wilso..