No abstract

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), 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)=5 for all addressings of the Petersen graph and then give an optimal addressing by 6-tuples.

Abstract. Weighted generalizations of Hoffman's ratio bound on the independence number of a regular graph are surveyed. Several known bounds are reviewed as special cases of modest extensions. Comparisons are made with the Shannon capacity Θ, Lovász' parameter ϑ, Schrijver's parameter ϑ ′ , and the ultimate independence ratio for categorical products. The survey concludes with some observations on graphs that attain a weighted version of a bound of Cvetković.

In this paper, we discuss various connections between the smallest eigenvalue of the adjacency matrix of a graph and its structure. There are several techniques for obtaining upper bounds on the smallest eigenvalue, and some of them are based on Rayleigh quotients, Cauchy interlacing using induced subgraphs, and Haemers interlacing with vertex partitions and quotient matrices. In this paper, we are interested in obtaining lower bounds for the smallest eigenvalue. Motivated by results on line graphs and generalized line graphs, we show how graph decompositions can be used to obtain such lower bounds.A 3 = rA + s(J − I) + tI for some nonnegative integers r, s, t depending on n, k, a, c. ThusThus, z is at least as large as the minimum root of the cubic x 3 − rx + s − t. Two of the roots are θ and τ , inherited from (5), and the other is necessarily −(θ + τ ) = c − a since the coefficient of x 2 is 0. Thus, λ(G) = z ≥ min{c − a, τ }. Therefore, taking G (3) in Theorem 2.1 gives λ(G) = τ when G is a strongly regular graph such that τ ≤ c − a. In particular, λ(G) = τ when a ≤ c, a condition that must be satisfied by at least one of a strongly regular graph and its complement.

Graham and Pollak showed that the vertices of any connected graph G can be assigned t-tuples with entries in {0, a, b}, called addresses, such that the distance in G between any two vertices equals the number of positions in their addresses where one of the addresses equals a and the other equals b. In this paper, we are interested in determining the minimum value of such t for various families of graphs. We develop two ways to obtain this value for the Hamming graphs and present a lower bound for the triangular graphs.