Greyson C. Wesley

Greyson C. Wesley

Mathematics PhD student

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publications

Enumerating Threshold Graphs and Some Related Graph Classes

Published in Journal of Integer Sequences, 2022

We give combinatorial proofs of some enumeration formulas involving labelled threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs. In each case we count by number of vertices and number of components. For threshold graphs, we also count by number of dominating vertices, and for loop-threshold graphs we count by number of looped dominating vertices. We also obtain an analog of the Frobenius formula (connecting Eulerian numbers and Stirling numbers of the second kind) in the context of labelled threshold graphs.

Recommended citation: David Galvin, Greyson Wesley, and Bailee Zacovic. "Enumerating Threshold Graphs and Some Related Graph Classes". Journal of Integer Sequences. 25 (2022), Article 22.2.7.
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\(k\)-NIM Trees: A Characterization and Enumeration

arXiv preprint, 2022

Among those real symmetric matrices whose graph is a given tree \(T\), the maximum multiplicity \(M(T)\) that can be attained by an eigenvalue is known to be the path cover number of \(T\). We say that a tree is \(k\)-NIM if, whenever an eigenvalue attains a multiplicity of \(k-1\) less than the maximum multiplicity, all other multiplicities are 1. 1-NIM trees are known as NIM trees, and a characterization for NIM trees is already known. Here we provide a graph-theoretic characterization for \(k\)-NIM trees for each \(k \geq 1\), as well as count them. It follows from the characterization that \(k\)-NIM trees exist on \(n\) vertices only when \(k=1,2,3\). In case \(k=3\), the only 3 -NIM trees are simple stars.

Recommended citation: C. R. Johnson, G. Tsoukalas, G. Wesley, and Z. Zhao. "\(k\)-NIM Trees: A Characterization and Enumeration" 2022. Preprint.
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Counting Monic Polynomials Without Prescribed Factors Over \(\mathbb{F}_q\)

Local manuscript, 2022

In this short note we use the symbolic method to provide a brief proof for the ordinary generating function of the combinatorial class of monic polynomials over \(\mathbb{F}_q\) not divisible by anything from a given set of monic irreducibles.

Recommended citation: G. Wesley. "Counting Monic Polynomials Without Prescribed Factors Over \(\mathbb{F}_q\)" (2022). Local manuscript.
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The Inverse Characteristic Polynomial Problem for Graphs over Finite Fields

Published in Recent Research in Polynomials, 2023

Let \(\mathbb{F}\) be a finite field, and let \(G\) be a graph on \(n\) vertices. We study the possible characteristic polynomials that may be realized by matrices \(A\) over a finite field such that the graph of \(A\) is \(G\). We focus mainly on the case \(G\) is a tree \(T\), not only because trees are computationally simpler, but also because the theory of eigenvalue multiplicities is much better understood for trees than it is for general graphs. We demonstrate the applications to this problem by branch duplication and the recently developed geometric Parter–Wiener, etc. theory. We end with a list of several conjectures which should pave the way for future study.

Recommended citation: C. Johnson, X. Lin, X. Liu, G. Wesley, and S. Zhou. "The Inverse Characteristic Polynomial Problem for Graphs over Finite Fields". In: F. Ozger (Ed.), Recent Research in Polynomials (2023), Chapter 6.
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