  In [1] Peters and Regts confirmed a conjecture by Sokal by showing that for
every $\Delta \in \mathbb{Z}_{\geq 3}$ 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.
