Maximizing H-colorings of a regular graph

For graphs $G$ and $H$, a {\em homomorphism} from $G$ to $H$, or {\em $H$-coloring} of $G$, is an adjacency preserving map from the vertex set of $G$ to the vertex set of $H$. Writing ${\rm hom}(G,H)$ for the number of $H$-colorings admitted by $G$, we conjecture that for any simple finite graph $H$ (perhaps with loops) and any simple finite $n$-vertex, $d$-regular, loopless graph $G$ we have $${\rm hom}(G,H) \leq \max{{\rm hom}(K_{d,d},H)^{\frac{n}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{n}{d+1}}}$$ where $K_{d,d}$ is the complete bipartite graph with $d$ vertices in each partition class, and $K_{d+1}$ is the complete graph on $d+1$ vertices. Results of Zhao confirm this conjecture for some choices of $H$ for which the maximum is achieved by ${\rm hom}(K_{d,d},H)^{n/2d}$. Here we exhibit infinitely many non-trivial triples $(n,d,H)$ for which the conjecture is true and for which the maximum is achieved by ${\rm hom}(K_{d+1},H)^{n/(d+1)}$. We also give sharp estimates for ${\rm hom}(K_{d,d},H)$ and ${\rm hom}(K_{d+1},H)$ in terms of some structural parameters of $H$. This allows us to characterize those $H$ for which ${\rm hom}(K_{d,d},H)^{1/2d}$ is eventually (for all sufficiently large $d$) larger than ${\rm hom}(K_{d+1},H)^{1/(d+1)}$ and those for which it is eventually smaller, and to show that this dichotomy covers all non-trivial $H$. Our estimates also allow us to obtain asymptotic evidence for the conjecture in the following form. For fixed $H$, for all $d$-regular $G$ we have $${\rm hom}(G,H)^{\frac{1}{|V(G)|}} \leq (1+o(1))\max{{\rm hom}(K_{d,d},H)^{\frac{1}{2d}}, {\rm hom}(K_{d+1},H)^{\frac{1}{d+1}}}$$ where $o(1)\rightarrow 0$ as $d \rightarrow \infty$. More precise results are obtained in some special cases.