On Unimodality of Independence Polynomials of Trees

TL;DR

This paper investigates the unimodality of independence polynomials in trees, providing computational evidence up to 20 vertices and establishing their log-concavity, which implies unimodality.

Contribution

It offers computational support for the conjecture that all trees have unimodal independence polynomials and proves their log-concavity.

Findings

Independence polynomials of trees with up to 20 vertices are unimodal.

These polynomials are shown to be log-concave, ensuring unimodality.

The study uses a database of non-isomorphic trees to efficiently compute independence polynomials.

Abstract

An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $\alpha{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for the sequence of numbers of independent sets of each size. In other words, the $k$-th coefficient of the independence polynomial equals the number of independent sets comprised of $k$ vertices. For instance, the degree of the independence polynomial of the graph $G$ is equal to $\alpha{(G)}$. In 1987, Alavi, Malde, Schwenk, and Erd{\"o}s conjectured that the independence polynomial of a tree is unimodal. In what follows, we provide support to this assertion considering trees with up to $20$ vertices. Moreover, we show that the corresponding independence polynomials are log-concave and, consequently, unimodal. The algorithm computing the independence polynomial of a given tree makes use of a database of non-isomorphic unlabeled trees to prevent repeated computations.