On the unimodality of independence polynomials of some graphs

TL;DR

This paper investigates the unimodality, log-concavity, and zero distribution of independence polynomials in graphs, providing recurrence relations, factorizations, and resolving some existing conjectures.

Contribution

It introduces new recurrence relations and factorizations for independence polynomials, settling several open unimodality conjectures for specific graph classes.

Findings

Established recurrence relations for independence polynomials

Provided factorizations for certain graph classes

Resolved some open unimodality conjectures

Abstract

In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems.