On a Conjecture of Sokal Concerning Roots of the Independence Polynomial

Abstract

A conjecture of Sokal [24], regarding the domain of nonvanishing for independence polynomials of graphs, states that given any natural number $\Delta \ge 3$, there exists a neighborhood in $\mathbb{C}$ of the interval $[0,(\Delta -1{)}^{\Delta -1}/(\Delta -2{)}^{\Delta })$ on which the independence polynomial of any graph with maximum degree at most $\Delta$ does not vanish. We show here that Sokal’s conjecture holds, as well as a multivariate version, and prove the optimality for the domain of nonvanishing. An important step is to translate the setting to the language of complex dynamical systems.