A conjecture of Sokal , regarding the domain of nonvanishing for independence polynomials of graphs, states that given any natural number , there exists a neighborhood in of the interval 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 the optimality for the domain of nonvanishing. An important step is to translate the setting to the language of complex dynamical systems.
"On a Conjecture of Sokal Concerning Roots of the Independence Polynomial." Michigan Math. J. 68 (1) 33 - 55, April 2019. https://doi.org/10.1307/mmj/1541667626