On derivatives and higher-order derivatives of chromatic polynomials

Let $G$ be a graph of order $n$ with maximum degree $\mathrm{\Delta }$, and let $P(G,x)$ denote its chromatic polynomial. We investigate several properties of $P(G,x)$ related to its derivatives and higher-order derivatives. First, we study the monotonicity of $P(G,x)\mathrm{/}{x}^n$. Dong proved that $(x-1{)}^{n}P(G,x)\ge x^{n}P(G,x-1)$ for all real $x\ge n$. In particular, taking $x=n$ establishes the Bartels-Welsh ``shameful conjecture" that $P(G,n)\mathrm{/}P(G,n-1)>e$. Fadnavis later showed that the same inequality holds for all real $x\ge 36{\mathrm{\Delta }}^{3\mathrm{/}2}$. We improve this bound by proving that it also holds for all real $x\ge 10{\mathrm{\Delta }}^{3\mathrm{/}2}$. We then consider a conjecture of Dong, Ge, Gong, Ning, Ouyang, and Tay asserting that $\frac{d^k}{d{x}^k}(\ln [(-1{)}^{n}P(G,x)])<0$ for all $k\ge 2$ and $x\in (-\mathrm{\infty },0)$. We establish this conjecture for all $k\ge 2$ and $x\le -3.01\mathrm{\Delta }k$.