  For a graph $\Gamma$, let $K(H_{\Gamma},1)$ denote the Eilenberg-Mac Lane
space associated to the right-angled Artin (RAA) group $H_{\Gamma}$ defined by
$\Gamma$. We use the relationship between the combinatorics of $\Gamma$ and the
topological complexity of $K(H_{\Gamma},1)$ to explain, and generalize to the
higher TC realm, Dranishnikov's observation that the topological complexity of
a covering space can be larger than that of the base space. In the process, for
any positive integer $n$, we construct a graph $\mathcal{O}_n$ whose
TC-generating function has polynomial numerator of degree $n$. Additionally,
motivated by the fact that $K(H_{\Gamma},1)$ can be realized as a polyhedral
product, we study the LS category and topological complexity of more general
polyhedral product spaces. In particular, we use the concept of a strong axial
map in order to give an estimate, sharp in a number of cases, of the
topological complexity of a polyhedral product whose factors are real
projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present
in the case of RAA groups.
