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We characterize the inertia of A + B for Hermitian matrices A and B when the rank of B is one. We use this to characterize the inertia of a partial join of two graphs. We then provide graph joins G for which the minimum number of complete bipartite graphs needed in a partition of the edge multi-set of G is equal to the maximum of the number of positive and… (More)

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… (More)

If G is a graph on n vertices and r 2 2, w e let m,(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, f(G). In determining m,(G), w e may assume that no two vertices of G have the same neighbor set. For such reduced graphs G, w e prove that m,(G) 2 log,(n + r-l)/r. Furthermore, for each… (More)

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… (More)

We consider the minimum number of cliques needed to partition the edge set of D(G), the distance multigraph of a simple graph G. Equivalently, we seek to minimize the number of elements needed to label the vertices of a simple graph G by sets so that the distance between two vertices equals the cardinality of the intersection of their labels. We use a… (More)

Let bp(+K v) be the minimum number of complete bipartite subgraphs needed to partition the edge set of +K v , the complete multigraph with + edges between each pair of its v vertices. Many papers have examined bp(+K v) for v2+. For each + and v with v2+, it is shown here that if certain Hadamard and conference matrices exist, then bp(+K v) must be one of… (More)

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