Some results on the maximal chromatic polynomials of $2$-connected $k$-chromatic graphs

In 2015, Brown and Erey conjectured that every $2$-connected graph $G$ on $n$ vertices with chromatic number $k\ge 4$ has at most $(x-1{)}_{k-1}((x-1{)}^{n-k+1}+(-1{)}^{n-k})$ proper $x$-colorings for all $x\ge k$. Engbers, Erey, Fox, and He proved this conjecture for $x=k$. In this paper, we prove Brown and Erey’s conjecture under the condition that either the clique number of $G$ is $k$, or the independent number of $G$ is $2$.