A note on the alternating number of independent sets in a graph

TL;DR

This paper improves a bound on the absolute value of the independence polynomial evaluated at -1, relating it to the minimum vertices needed to eliminate certain cycles, with implications in mathematics and physics.

Contribution

It introduces a tighter bound based on the removal of vertices to eliminate induced cycles divisible by 3, refining previous results by Engström.

Findings

New bound: |I(G;-1)| ≤ 2^{φ_3(G)}

Bound is sharp and attainable by some connected graphs

Enhances understanding of independence polynomial behavior

Abstract

The independence polynomial of a graph $G$ evaluated at $-1$, denoted here as $I(G;-1)$, has arisen in a variety of different areas of mathematics and theoretical physics as an object of interest. Engstr\"om used discrete Morse theory to prove that $\left|I(G;-1)\right|\leq 2^{\phi(G)}$ where $\phi(G)$ is the decycling number of $G$, i.e., the minimum number of vertices needed to be deleted from $G$ so that the remaining graph is acyclic. Here, we improve Engstr\"om's bound by showing $\left|I(G;-1)\right|\leq 2^{\phi_3(G)}$ where $\phi_3(G)$ is the minimum number of vertices needed to be deleted from $G$ so that the resulting graph contains no induced cycles whose length is divisible by $3$. We also note that this bound is not just sharp but that every value in the range given by the bound is attainable by some connected graph.