Symmetric and unimodal independence polynomials of trees

TL;DR

This paper investigates the existence of trees with independence polynomials that are both symmetric and unimodal, focusing on their degree and vertex count.

Contribution

It explores conditions under which trees have symmetric and unimodal independence polynomials, advancing understanding of polynomial properties in graph theory.

Findings

Identifies when trees have symmetric and unimodal independence polynomials.

Establishes relationships between tree size and polynomial symmetry and unimodality.

Provides conditions for the existence of such polynomials of specific degrees.

Abstract

Given $n \geq 1$, we study the existence of a tree on $n$ vertices whose independence polynomial is symmetric and unimodal as well as the existence of a symmetric and unimodal independence polynomial of degree $n$ of a tree.