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

The purpose of the present paper is to provide, for all pairs of integers $(\mathrm{\Delta },g)$ with $\text{D}\ge 3$ and $g\ge 3$, a positive number $C(\mathrm{\Delta },g)$ such that chromatic polynomial $P_G(q)$ of a graph $G$ with maximum degree $\mathrm{\Delta }$ and finite girth $g$ is free of zero if $\mathrm{\mid }q\mathrm{\mid }\ge C(\mathrm{\Delta },g)$. 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 $\text{D}$ 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 $\mathrm{\Delta }\to \mathrm{\infty }$.