On The Roots of Independence Polynomial: Quantifying The Gap

The independence polynomial of a graph $G$ is the generating polynomial corresponding to its independent sets of different sizes. More formally, if $a_k(G)$ denotes the number of independent sets of $G$ of size $k$ then $$I(G,z) \as \sum_{k}^{} (-1)^k a_k(G) z^k.$$ The study of evaluating $I(G,z)$ has several deep connections to problems in combinatorics, complexity theory and statistical physics. Consequently, the roots of the independence polynomial have been studied in detail. In particular, many works have provided regions in the complex plane that are devoid of any roots of the polynomial. One of the first such results showed a lower bound on the absolute value of the smallest root $\beta(G)$ of the polynomial. Furthermore, when $G$ is connected, Goldwurm and Santini established that $\beta(G)$ is a simple real root of $I(G,z)$ smaller than one. An alternative proof was given by Csikv\'ari. Both proofs do not provide a gap from $\beta(G)$ to the smallest absolute value amongst all the other roots of $I(G,z)$. In this paper, we quantify this gap.