Addressing the Petersen graph

Motivated by a problem on message routing in communication networks, Graham and Pollak proposed 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 observed that N⩾h(D)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 of Gregory, Shader and Watts yields a necessary condition for equality to occur. As an illustration, we show that N>h(D)=5N>h(D)=5 for all addressings of the Petersen graph and then give an optimal addressing by 6-tuples.