In [PR19], Peters and Regts confirmed a conjecture by Sokal [Sok01] by showing that for every , there exists a complex neighborhood of the interval on which the independence polynomial is nonzero for all graphs of maximum degree Δ. Furthermore, they gave an explicit neighborhood containing this interval on which the independence polynomial is nonzero for all finite rooted Cayley trees with branching number Δ. The question remained whether would be zero-free for the independence polynomial of all graphs of maximum degree Δ. In this paper, we show that this is not the case.
"Cayley Trees do Not Determine the Maximal Zero-Free Locus of the Independence Polynomial." Michigan Math. J. 70 (3) 635 - 648, August 2021. https://doi.org/10.1307/mmj/1599206419