  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.
