On the zero-free region for the chromatic polynomial of graphs with maximum degree $\Delta$ and girth $g$

The purpose of the present paper is to provide, for all pairs of integers $(\Delta,g)$ with $\D\ge 3$ and $g\ge 3$, a positive number $C(\Delta, g)$ such that chromatic polynomial $P_G(q)$ of a graph $G$ with maximum degree $\Delta$ and finite girth $g$ is free of zero if $|q|\ge C(\Delta, g)$. Our bounds enlarge the zero-free region in the complex plane of $P_G(q)$ in comparison to previous bounds. In particular, for small values of $\D$ our estimates yield a sensible improvement on the bounds recently obtained by Jenssen, Patel and Regts in \cite{JPR}, while they coincide with those of \cite{JPR} when $\Delta\to \infty$.