  The chromatic polynomial is a well studied object in graph theory. There are
many results and conjectures about the log-concavity of the chromatic
polynomial and other polynomials related to it. The location of the roots of
these polynomials has also been well studied. One famous result due to A. Sokal
and C. Borgs provides a bound on the absolute value of the roots of the
chromatic polynomial in terms of the highest degree of the graph. We use this
result to prove a modification of a log-concavity conjecture due to F. Brenti.
The original conjecture of Brenti was that the chromatic polynomial is
log-concave on the natural numbers. This was disproved by Paul Seymour by
presenting a counter example. We show that the chromatic polynomial $P_G(q)$ of
graph $G$ is in fact log-concave for all $q > C\Delta + 1$ for an explicit
constant $C < 10$, where $\Delta$ denotes the highest degree of $G$. We also
provide an example which shows that the result is not true for constants $C$
smaller than 1.
