In this famous paper, June Huh proved Read's conjecture, which claims that coefficients of a characteristic polynomial $\chi_{G}(q) = a_{n}q^{n} - a_{n-1}q^{n-1} + \dots + (-1)^{n}a_{0}$ of any graph satisfy log-concave property, i.e. $$ a_{i-1}a_{i+1}\leq a_{i}^{2} $$ for any $i$. My question is - is the converse true? For any given polynomial whose coefficients satisfies the above equation, can we find a graph whose chromatic polynomial is exactly same as the given polynomial? I believe that this conjecture is false, but I can't find any counterexample (actually I don't know how to find such an example).
I considered this question while I was thinking about the following $\alpha + n$ conjecture:
For any algebraic integer $\alpha$, there exists $n\in \mathbb{Z}_{>0}$ such that $\alpha+n$ is a root of some chromatic polynomial.
To disprove the conjecture, it is enough to find an algebraic integer $\alpha$ such that $\alpha + n$ can't be a root of any polynomial that satisfies the log-concave condition. But after some computation, I found that it is impossible to find such $\alpha$. In fact, for any $p(x) = x^{d} + b_{d-1}x^{d-1} + \cdots + b_{0}\in\mathbb{Z}[x]$, $i$-th coefficient of the polynomial $p(x-n)$ is $$ c_{i}(n) = (-1)^{d-i}\left(\binom{d}{i}n^{d-i} - \binom{d}{i+1}n^{d-i-1}b_{d-1} + \binom{d}{i+2}n^{d-i-2}b_{d-2} - \cdots \right) $$ and then $c_{i-1}(n)c_{i+1}(n)\leq c_{i}(n)^{2}$ for sufficiently large $n$ because binomial coefficients $\binom{d}{0}, \binom{d}{1}, \dots, \binom{d}{d}$ is strictly log-concave. Hence we need another approach (completely different approach, if we believe that the conjecture is true!). FYI, here is a proof of the conjecture when $\alpha$ is a quadratic integer.
Well, $K_3$ has chromatic polynomial $x^3-3x^2+2x$.
$x^3-3000x^2+2x$ is still log-concave, but it cannot be the chromatic polynomial of any graph $G$ since that would mean that the number of ways to color $G$ with a small number of colors is negative. I don't think it is very hard to come up with higher degree counterexamples.