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arXiv · 1701.08049

On a conjecture of Sokal concerning roots of the independence polynomial

Han Peters·Guus Regts
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Abstract

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

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Han Peters, Guus Regts. 2018-06-22. On a conjecture of Sokal concerning roots of the independence polynomial. https://arxiv.org/abs/1701.08049

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