Almost Unimodal and Real-Rooted Graph Polynomials

TL;DR

This paper investigates conditions under which certain graph polynomials are unimodal or real-rooted, providing new theorems that connect graph properties with polynomial root characteristics, extending known results in graph polynomial theory.

Contribution

It introduces new theorems linking hereditary and complement properties of graphs to the unimodality and real-rootedness of associated graph polynomials.

Findings

Unimodality of coefficients for almost all graphs with certain properties.

Real-rootedness of specific graph polynomials characterized by graph properties.

Extension of known results to broader classes of graph polynomials.

Abstract

It is well known that the coefficients of the matching polynomial are unimodal. Unimodality of the coefficients (or their absolute values) of other graph polynomials have been studied as well. One way to prove unimodality is to prove real-rootedness.` Recently I. Beaton and J. Brown (2020) proved the for almost all graphs the coefficients of the domination polynomial form a unimodal sequence, and C. Barton, J. Brown and D. Pike (2020) proved that the forest polynomial (aka acyclic polynomial) is real-rooted iff $G$ is a forest. Let $\mathcal{A}$ be a graph property, and let $a_i(G)$ be the number of induced subgraphs of order $i$ of a graph $G$ which are in $\mathcal{A}$. Inspired by their results we prove: {\bf Theorem:} If $\mathcal{A}$ is the complement of a hereditary property, then for almost all graphs in $G(n,p)$ the sequence $a_i(G)$ is unimodal. {\bf Theorem:} If $\mathcal{A}$ is a hereditary property which contains a graph which is not a clique or the complement of a clique, then the graph polynomial $P_{\mathcal{A}}(G;x) = \sum_i a_i(G) x^i$ is real-rooted iff $G \in \mathcal{A}$.