On a conjecture of Sokal concerning roots of the independence polynomial

A conjecture of Sokal [22] regarding the domain of non-vanishing for independence polynomials of graphs, states that given any natural number ∆ ≥ 3, there exists a neighborhood in C of the interval [0, (∆−1) ∆−1 (∆−2)∆ ) on which the independence polynomial of any graph with maximum degree at most ∆ does not vanish. We show here that Sokal’s Conjecture holds, as well as a multivariate version, and prove optimality for the domain of non-vanishing. An important step is to translate the setting to the language of complex dynamical systems.