Kevin N. Vander Meulen

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A sign pattern Z (a matrix whose entries are elements of {+, −, 0}) is spectrally arbitrary if for any self-conjugate spectrum there is a real matrix with sign pattern Z having the given spectrum. Spectrally arbitrary sign patterns were introduced in [5], where it was (incorrectly) stated that if a sign pattern Z is reducible and each of its irreducible(More)
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)
A new family of minimal spectrally arbitrary patterns is presented which allow for arbitrary spectrum by using the Nilpotent-Jacobian method introduced in [J. The novel approach here is the use of the Intermediate Value Theorem to avoid finding an explicit nilpotent realization of the new minimal spectrally arbitrary patterns. 1. Introduction. A matrix S(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)
Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper an investigation is initiated into which zero-nonzero patterns are potentially nilpotent over F with a special emphasis on the case that F = Zp is a finite field. A(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)