The independence polynomial of trees is not always log-concave starting from order 26

TL;DR

This paper demonstrates that the independence polynomial of trees, previously conjectured to be log-concave, can actually fail to be so starting from trees of order 26, providing counterexamples to a long-standing belief.

Contribution

The authors construct infinite families of trees with independence polynomials that are not log-concave, disproving the conjecture for trees of sufficiently large order.

Findings

Independence polynomials of certain trees are not log-concave.

Counterexamples exist for trees of order 26 and above.

The conjecture of log-concavity for all trees is false.

Abstract

An independent set in a graph is a collection of vertices that are not adjacent to each other. The cardinality of the largest independent set in $G$ is represented by $\alpha(G)$. The independence polynomial of a graph $G = (V, E)$ was introduced by Gutman and Harary in 1983 and is defined as \[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \] where $s_k$ represents the number of independent sets in $G$ of size $k$. The conjecture made by Alavi, Malde, Schwenk, and Erd\"os in 1987 stated that the independence polynomials of trees are unimodal, and many researchers believed that this conjecture could be strengthened up to its corresponding log-concave version. However, in our paper, we present evidence that contradicts this assumption by introducing infinite families of trees whose independence polynomials are not log-concave.