Improved bounds for zeros of the chromatic polynomial on bounded degree graphs

We prove that for any graph $G$ of maximum degree at most $\mathrm{\Delta }$, the zeros of its chromatic polynomial ${\chi }_G(z)$ (in $\mathbb{C}$) lie outside the disk of radius $5.02\mathrm{\Delta }$ centered at $0$. This improves on the previously best known bound of approximately $6.91\mathrm{\Delta }$. In the case of graphs of high girth we can improve this. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $\mathrm{\Delta }$ and girth at least $g$, the zeros of its chromatic polynomial ${\chi }_G(z)$ lie outside the disk of radius $K_g\mathrm{\Delta }$ centered at $0$ where $K_g\to 1+e\approx 3.72$ as $g\to \mathrm{\infty }$. Finally, we give improved bounds on the Fisher zeros of the partition function of the Ising model.