Kevin N. Vander Meulen

Inertia and biclique decompositions of joins of graphs

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 negative Eigenvalues of G.

ON DETERMINING MINIMAL SPECTRALLY ARBITRARY PATTERNS

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. 308:121- 137, 2000). 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.

INERTIALLY ARBITRARY NONZERO PATTERNS OF ORDER 4

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 is inertially arbitrary but has no signing that is inertially arbitrary. In fact, up to equivalence, this pattern is unique among the irreducible order 4 patterns with this property.

Independence Complexes of Well-Covered Circulant Graphs

ABSTRACT We study the independence complexes of families of well-covered circulant graphs discovered by Boros–Gurvich–Milanič, Brown–Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g., vertex decomposable, shellable) or topological (e.g., Cohen–Macaulay, Buchsbaum) properties. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen–Macaulay, and/or Buchsbaum. A highlight of this search is an example of a graph whose independence complex is shellable but not vertex decomposable.

Cohen–Macaulay Circulant Graphs

Let G be the circulant graph C n (S) with , and let I(G) denote the edge ideal in the ring R = k[x 1,…, x n ]. We consider the problem of determining when G is Cohen–Macaulay, i.e, R/I(G) is a Cohen–Macaulay ring. Because a Cohen–Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C n (1, 2,…, d). We also characterize which cubic circulant graphs are Cohen–Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen–Macaulay property.

Sharp bounds for decompositions of graphs into complete r -partite subgraphs

If G is a graph on n vertices and r ≥ 2, we let mr(G) denote the minimum number of complete multipartite subgraphs, with r or fewer parts, needed to partition the edge set, E(G). In determining mr(G), we may assume that no two vertices of G have the same neighbor set. For such reducedgraphs G, we prove that mr(G) ≥ log2 (n + r − 1)/r. Furthermore, for each k ≥ 0 and r ≥ 2, there is a unique reduced graph G = G(r, k) with mr(G) = k for which equality holds. We conclude with a short proof of the known eigenvalue bound mr(G) ≥ max{n+ (G, n−(G)/(r − 1)}, and show that equality holds if G = G(r, k).

Decompositions of Complete Multigraphs Related to Hadamard Matrices

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 itsvvertices. Many papers have examined bp(?Kv) forv?2?. For each?andvwithv?2?, it is shown here that if certain Hadamard and conference matrices exist, then bp(?Kv) must be one of two numbers. Also, generalizations to decompositions and covers by completes-partite subgraphs are discussed and connections to designs and codes are presented.

ZERO-NONZERO PATTERNS FOR NILPOTENT MATRICES OVER FINITE FIELDS ∗

Fix a field F. A zero-nonzero pattern A is said to be potentiallynilpotent 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 potentiallynilpotent over F with a special emphasis on the case that F = Zp is a finite field. A necessarycondition on F is observed for a pattern to be potentiallynilpotent when the associated digraph has m loops but no small k-cycles, 2 ≤ k ≤ m − 1. As part of this investigation, methods are developed, using the tools of algebraic geometryand commutative algebra, to eliminate zero-nonzero patterns A as being potentiallynilpotent over anyfield F. These techniques are then used to classifyall irreducible zero-nonzero patterns of order two and three that are potentiallynilpotent over Zp for each prime p.

Rank decompositions and signed bigraphs

A signed bipartite graph G with vertices 1′, 2′,…,m and 1′,2′,…n′, determines the family M(G) consisting of all m by n matrices whose (i,j)-entry is zero if i,j′ )is not and edge of G nonnegative if {i,j′} is an edge of G with label +1, and nonpositive if {i,f′}is an edge of G with label -1. we show that each matrix A in M(G) can be expressed as the sum of rank(A) rank on matrices in M(G) if and only if for every cycle γ of G of length l(γ)≥6. We also show that each matrix in M(G) has its rank equal to its term rank if and only if (1) holds for every cycle γ of G. Graphical characterizations of the signed bigraphs whose cycles satisfy (1) and of the signed bigraphs whose cycles of length 6 or more satisfy (1) are given.

Bounds on polynomial roots using intercyclic companion matrices

Abstract The Frobenius companion matrix, and more recently the Fiedler companion matrices, have been used to provide lower and upper bounds on the modulus of any root of a polynomial p ( x ) . In this paper we explore new bounds obtained from taking the 1-norm and ∞-norm of a matrix in the wider class of intercyclic companion matrices. As is the case with Fiedler matrices, we observe that the new bounds from intercyclic companion matrices can improve those from the Frobenius matrix by at most a factor of two. By using the Hessenberg form of an intercyclic companion matrix, we describe how to determine the best upper bound when restricted to Fiedler companion matrices using the ∞-norm. We also obtain a new general bound by considering the polynomial x q p ( x ) for q > 0 . We end by considering upper bounds obtained from inverses of monic reversal polynomials of intercyclic companion matrices, noting that these can make more significant improvements on the bounds from a Frobenius companion matrix for certain polynomials.