On a conjecture of Sokal concerning roots of the independence polynomial

A conjecture of Sokal (2001) regarding the domain of non-vanishing for independence polynomials of graphs, states that given any natural number $\Delta \ge 3$, there exists a neighborhood in $\mathbb C$ of the interval $[0, \frac{(\Delta-1)^{\Delta-1}}{(\Delta-2)^{\Delta}})$ 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 optimality for the domain of non-vanishing. An important step is to translate the setting to the language of complex dynamical systems.