Unimodality of the independence polynomials of some composite graphs

TL;DR

This paper investigates the unimodality and log-concavity of independence polynomials for certain composite graphs, providing conditions under which these properties hold, and constructs specific graphs with symmetric, real-zero polynomials.

Contribution

It establishes new criteria for unimodality and log-concavity of independence polynomials in composite graphs and constructs graphs with specific polynomial properties addressing an open problem.

Findings

Conditions for unimodality of independence polynomials in lexicographic products.

Criteria for log-concavity of independence polynomials in composite graphs.

Existence of connected non-tree graphs with symmetric, real-zero independence polynomials.

Abstract

Let $I(G;x)$ denote the independence polynomial of a graph $G$. In this paper we study the unimodality properties of $I(G;x)$ for some composite graphs $G$. Given two graphs $G_1$ and $G_2$, let $G_1[G_2]$ denote the lexicographic product of $G_1$ and $G_2$. Assume $I(G_1;x)=\sum_{i\geq0}a_ix^i$ and $I(G_2;x)=\sum_{i\geq0}b_ix^i$, where $I(G_2;x)$ is log-concave. Then we prove (i) if $I(G_1;x)$ is log-concave and $(a^2_i-a_{i-1}a_{i+1})b^2_1\geq a_ia_{i-1}b_2$ for all $1\leq i \leq \alpha(G_1)$, then $I(G_1[G_2];x)$ is log-concave; (ii) if $a_{i-1}\leq b_1a_i$ for $1\leq i\leq \alpha(G_1)$, then $I(G_1[G_2];x)$ is unimodal. In particular, if $a_i$ is increasing in $i$, then $I(G_1[G_2];x)$ is unimodal. We also give two sufficient conditions when the independence polynomial of a complete multipartite graph is unimodal or log-concave. Finally, for every odd positive integer $\alpha > 3$, we find a connected graph $G$ not a tree, such that $\alpha(G) =\alpha$, and $I(G; x)$ is symmetric and has only real zeros. This answers a problem of Mandrescu and Miric\u{a}.