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 selfconjugate 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)
If G is a graph on n vertices and r 2 2, we 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), we 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 k(More)
A sign pattern is a matrix with entries in {+,−, 0}. A full sign pattern has no zero entries. The refined inertia of a matrix pattern is defined and techniques are developed for constructing potentially nilpotent full sign patterns. Such patterns are spectrally arbitrary. These techniques can also be used to construct potentially nilpotent sign patterns(More)
Throughout the paper, G denotes a graph that has no loops, but that may have multiple edges. The multi-set of edges of G is denoted by E(G). A graph G is simple if it has no (loops or) multiple edges. A biclique of G is a simple complete bipartite subgraph of G. A biclique decomposition of G is a collection of bicliques of G, such that each edge of G is in(More)
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(More)
A sign pattern Z (a matrix whose entries are elements of {+,−, 0}) is spectrally arbitrary if for any selfconjugate 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)
Inertially arbitrary nonzero patterns of order at most 4 are characterized. Some of these patterns are demonstrated to be inertially arbitrary but not spectrally arbitrary. The order 4 sign patterns which are inertially arbitrary and have a nonzero pattern that is not spectrally arbitrary are also described. There exists an irreducible nonzero pattern which(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.H. Drew, C.R. Johnson, D.D. Olesky, and P. van den Driessche. Spectrally arbitrary patterns.Lin. Alg. and Appl. 308:121137, 2000]. The novel approach here is the use of the Intermediate Value Theorem(More)
Let bp(+Kv) be the minimum number of complete bipartite subgraphs needed to partition the edge set of +Kv , the complete multigraph with + edges between each pair of its v vertices. Many papers have examined bp(+Kv) for v 2+. For each + and v with v 2+, it is shown here that if certain Hadamard and conference matrices exist, then bp(+Kv) must be one of two(More)