On the location of roots of the independence polynomial of bounded degree graphs

Abstract

In [1] Peters and Regts confirmed a conjecture by Sokal by showing that for every $\Delta \in {\mathbb{Z}}_{\ge 3}$ there exists a complex neighborhood of the interval $\left[0,\frac{{\left(\Delta -1\right)}^{\Delta -1}}{{\left(\Delta -2\right)}^{\Delta }}\right)$ on which the independence polynomial is nonzero for all graphs of maximum degree $\Delta$. Furthermore, they gave an explicit neighborhood $U_{\Delta }$ containing this interval on which the independence polynomial is nonzero for all finite rooted Cayley trees with branching number $\Delta$. The question remained whether $U_{\Delta }$ would be zero-free for the independence polynomial of all graphs of maximum degree $\Delta$. In this paper it is shown that this is not the case. [1] Han Peters and Guus Regts, On a conjecture of sokal concerning roots of the independence polynomial, Michigan Math. J. (2019), Advance publication.

11 pages, 1 figure