Cayley Trees do Not Determine the Maximal Zero-Free Locus of the Independence Polynomial

In [PR19], Peters and Regts confirmed a conjecture by Sokal [Sok01] by showing that for every Δ∈Z_≥3, there exists a complex neighborhood of the interval [0,(Δ−1)^Δ−1/(Δ−2) Δ) on which the independence polynomial is nonzero for all graphs of maximum degree Δ. Furthermore, they gave an explicit neighborhood U_Δ containing this interval on which the independence polynomial is nonzero for all finite rooted Cayley trees with branching number Δ. The question remained whether U_Δ 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.