Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Fall 2022

Speaker:  Rob Davis, Colgate University
Title: Perfectly Matchable Set Polynomials and an Application to Ehrhart Theory
Time: 2:30 PM, Monday, September 12, 2022
Place:  Malott 206

Abstract: A subset $S$ of vertices of a graph $G$ is called a perfectly matchable set of $G$ if the subgraph induced by $S$ contains a perfect matching. The perfectly matchable set polynomial of $G$, first made explicit by Ohsugi and Tsuchiya, is the (ordinary) generating function $p(G; z)$ for the number of perfectly matchable sets of $G$. In this talk, we will compare $p(G; z)$ to the classical matching polynomial and provide explicit recurrences for computing $p(G; z)$ for an arbitrary (simple) graph. We will use these to compute the Ehrhart $h^\ast$-polynomials for certain lattice polytopes, which was the original motivation for this work. Namely, we show that $p(G; z)$ is the $h^\ast$-polynomial for certain classes of stable set polytopes, whose vertices correspond to stable sets of $G$. This is joint work with Florian Kohl.

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