| Abstract |
A conjecture of Sokal [24], regarding the domain of nonvanishing for independence polynomials of graphs, states that given any natural number Δ≥3, there exists a neighborhood in C of the interval [0,(Δ−1)Δ−1/(Δ−2)Δ) on which the independence polynomial of any graph with maximum degree at most Δ 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.
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