A note on the values of independence polynomials at $-1$

TL;DR

This paper proves a conjecture that for any positive integer k and integer q with |q| ≤ 2^k, there exists a connected graph with decycling number k and independence polynomial evaluated at -1 equal to q.

Contribution

The paper confirms the conjecture that all integers within the bounds can be realized as the independence polynomial at -1 for connected graphs with a given decycling number.

Findings

Confirmed the conjecture for all positive integers k and integers q with |q| ≤ 2^k.

Established the existence of connected graphs with prescribed independence polynomial values at -1.

Extended understanding of the relationship between graph invariants and polynomial evaluations.

Abstract

The independence polynomial $I(G;x)$ of a graph $G$ is $I(G;x)=\sum_{k=1}^{\alpha(G)} s_k x^k$, where $s_k$ is the number of independent sets in $G$ of size $k$. The decycling number of a graph $G$, denoted $\phi(G)$, is the minimum size of a set $S\subseteq V(G)$ such that $G-S$ is acyclic. Engstr\"om proved that the independence polynomial satisfies $|I(G;-1)| \leq 2^{\phi(G)}$ for any graph $G$, and this bound is best possible. Levit and Mandrescu provided an elementary proof of the bound, and in addition conjectured that for every positive integer $k$ and integer $q$ with $|q|\leq 2^k$, there is a connected graph $G$ with $\phi(G)=k$ and $I(G;-1)=q$. In this note, we prove this conjecture.