arXiv subjects

Kevin N. Vander Meulen

Publications and source records attributed to Kevin N. Vander Meulen.

16 recordsLinked to original sources

arXiv2026

Kemeny's constant and Braess cliques in graphs

Kemeny's constant is used as a measure of the average travel time on a graph. Braess' paradox for graphs is the observation that in some graphs, when an edge is added, Kemeny's constant increases. We introduce the notion of a Braess clique $K_\ell$, a clique that when inserted into a graph on an independent set of $\ell$ vertices, will create an increase in Kemeny's constant. In this context, a Braess edge is a Braess $K_2$. We provide examples of graphs that have a Braess $K_\ell$ for $\ell\geq 3$. We observe that almost every connected planar labelled graph has a Braess $K_\ell$ for each $\ell\geq 3$. We also explore the relationship between Braess edges and Braess cliques in graphs.

math.CO↗

arXiv2025

Sign patterns which require or allow the strong multiplicity property

We initiate a study of sign patterns that require or allow the non-symmetric strong multiplicity property (nSMP). We show that all cycle patterns require the nSMP, regardless of the number of nonzero diagonal entries. We present a class of Hessenberg patterns that require the nSMP. We characterize which star sign patterns require, which allow, and which do not allow the nSMP. We show that if a pattern requires distinct eigenvalues, then it requires the nSMP. Further, we characterize the patterns that allow the nSMP as being precisely the set of patterns that allow distinct eigenvalues, a property that corresponds to a simple feature of the associated digraph. We also characterize the sign patterns of order at most three according to whether they require, allow, or do not allow the nSMP.

math.RA↗

arXiv2023

Condition Numbers of Hessenberg Companion Matrices

The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class of companion matrices that can be characterized with a Hessenberg form. In this paper, we demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices. We also show that there are other companion matrices which can provide a much smaller condition number than any Fiedler companion matrix. We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial.

math.RA↗

arXiv2020

Well-covered Token Graphs

The $k$-token graph $T_k(G)$ is the graph whose vertices are the $k$-subsets of vertices of a graph $G$, with two vertices of $T_k(G)$ adjacent if their symmetric difference is an edge of $G$. We explore when $T_k(G)$ is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs $G$, we classify when $T_k(G)$ is well-covered. For an arbitrary graph $G$, we show that if $T_2(G)$ is well-covered, then the girth of $G$ is at most four. We include upper and lower bounds on the independence number of $T_k(G)$, and provide some families of well-covered token graphs.

math.CO↗

arXiv2019

Maximum nullity and zero forcing of circulant graphs

It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of certain circulant graphs, including some bipartite circulants, cubic circulants, and circulants which are torus products, to obtain bounds on the minimum rank and the maximum nullity. We also evaluate when the zero forcing number will give equality.

math.CO↗

arXiv2018

Shedding vertices of vertex decomposable graphs

We focus our attention on well-covered graphs that are vertex decomposable. We show that for many known families of these vertex decomposable graphs, the set of shedding vertices forms a dominating set. We then construct three new infinite families of well-covered graphs, none of which have this property. We use these results to provide a minimal counterexample to a conjecture of Villarreal regarding Cohen-Macaulay graphs.

math.CO↗

arXiv2017

Bounds on polynomial roots using intercyclic companion matrices

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 $\infty$-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 $\infty$-norm. We also obtain a new general bound by considering the polynomial $x^qp(x)$ 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.

math.RA↗

arXiv2017

Bordering for spectrally arbitrary sign patterns

We develop a matrix bordering technique that can be applied to an irreducible spectrally arbitrary sign pattern to construct a higher order spectrally arbitrary sign pattern. This technique generalizes a recently developed triangle extension method. We describe recursive constructions of spectrally arbitrary patterns using our bordering technique, and show that a slight variation of this technique can be used to construct inertially arbitrary sign patterns.

math.RA↗

arXiv2016

Spectrally arbitrary pattern extensions

A matrix pattern is often either a sign pattern with entries in {0,+,-} or, more simply, a nonzero pattern with entries in {0,*}. A matrix pattern A is spectrally arbitrary if for any choice of a real matrix spectrum, there is a real matrix having the pattern A and the chosen spectrum. We describe a graphical technique, a triangle extension, for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns. These methods provide a new way of viewing some known spectrally arbitrary patterns, as well as providing many new families of spectrally arbitrary patterns. We also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns. We then provide an additional extension method for zero-nonzero patterns.

math.CO↗

arXiv2016

Refined Inertia of Matrix Patterns

We explore how the combinatorial arrangement of prescribed zeros in a matrix affects the possible eigenvalues that the matrix can obtain. We demonstrate that there are inertially arbitrary patterns having a digraph with no 2-cycle, unlike what happens for nonzero patterns. We develop a class of patterns that are refined inertially arbitrary but not spectrally arbitrary, making use of the property of a properly signed nest. We include a characterization of the inertially arbitrary and refined inertially arbitrary patterns of order three, as well as the patterns of order four with the least number of nonzero entries.

math.RA↗

arXiv2015

Shellability, vertex decomposability, and lexicographical products of graphs

We investigate when the independence complex of $G[H]$, the lexicographical product of two graphs $G$ and $H$, is either vertex decomposable or shellable. As an application, we construct an infinite family of graphs with the property that every graph in this family has the property that the independence complex of each graph is shellable, but not vertex decomposable.

math.CO↗

arXiv2015

Independence complexes of well-covered circulant graphs

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) structure. 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.

math.CO↗

arXiv2012

Cohen-Macaulay Circulant Graphs

Let G be the circulant graph C_n(S) with S a subset of {1,2,...,\lfloor n/2 \rfloor}, and let I(G) denote its 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.

math.AC↗

arXiv2010

Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method

A nonzero pattern is a matrix with entries in {0,*}. A pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern. We develop ways to construct some potentially nilpotent patterns, including some balanced tree patterns. We explore the index of some of the nilpotent matrices constructed,and observe that some of the balanced trees are spectrally arbitrary using the Nilpotent-Jacobian method. Inspired by an argument in [R. Pereira, Nilpotent matrices and spectrally arbitrary sign patterns. Electron. J. Linear Algebra, 16 (2007), 232--236], we also uncover a feature of the Nilpotent-Jacobian method. In particular, we show that if N is the nilpotent matrix employed in this method to show that a pattern is a spectrally arbitary pattern, then N must have full index.

math.RA↗

arXiv2008

Zero-nonzero patterns for nilpotent matrices over finite fields

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 we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Z_p is a finite field. As part of this investigation, we develop methods, using the tools of algebraic geometry and commutative algebra, to eliminate zero-nonzero patterns A as being potentially nilpotent over any field F. We then use these techniques to classify all irreducible zero-nonzero patterns of order two and three that are potentially nilpotent over Z_p for each prime p.

math.RA↗

arXiv2005

On Determining Minimal Spectrally Arbitrary Patterns

In this paper we present a new family of minimal spectrally arbitrary patterns which allow for arbitrary spectrum by using the Nilpotent-Jacobian method. The novel approach here is that we use the Intermediate Value Theorem to avoid finding an explicit nilpotent realization of the new minimal spectrally arbitrary patterns.

math.RA↗