Let $\chi_G(q)$ be the chromatic polynomial of a graph $G$ with $n$ vertices. (More generally, $\chi_{\mathcal{A}}(q)$ can be the characteristic polynomial of a finite hyperplane arrangement $\mathcal{A}$.) Write $\widetilde{\chi}_G(q)=(-1)^n\chi_G(-q)=\sum a_kq^k$, which is obtained by replacing each coefficient of $\chi_G(q)$ by its absolute value. We can think of $\widetilde{\chi}_G(q)$ as defining a probability distribution $p$ on $\{1,\dots,n\}$, with $p(k)=a_k/\widetilde{\chi}_G(1)$. (We always have $a_0=0$, except for the empty graph.) The mean $\mu$ of $p$ is given by $\mu=\widetilde{\chi}_G^\prime(1)/\widetilde{\chi}_G(1)$, and the mode $m$ by $a_m = \max_k a_k$. (Possibly $m$ is multivalued.)
I am wondering whether anything can be said in general about $m$. If all the zeros of $\widetilde{\chi}_G(q)$ are real, then by a theorem of Darroch (see for instance Exercise 1.48(a) of Enumerative Combinatorics, vol. 1, second ed.) $|\mu-m|<1$. To what extent does this result hold "in general"? Is it approximately true (where we need to define "approximately" precisely) for all graphs, or at least for "generic" graphs?
