  We construct a family of countexamples to a conjecture of Galvin [5], which
stated that for any $n$-vertex, $d$-regular graph $G$ and any graph $H$
(possibly with loops), \[\hom(G,H) \leq \max\left\lbrace\hom(K_{d,d},
H)^{\frac{n}{2d}}, \hom(K_{d+1},H)^{\frac{n}{d+1}}\right\rbrace,\] where
$\hom(G,H)$ is the number of homomorphisms from $G$ to $H$.
  By exploiting properties of the graph tensor product and graph
exponentiation, we also find new infinite families of $H$ for which the bound
stated above on $\hom(G,H)$ holds for all $n$-vertex, $d$-regular $G$.
  In particular we show that if $H_{\rm WR}$ is the complete looped path on
three vertices, also known as the Widom-Rowlinson graph, then $$
{\hom}(G,H_{\rm WR}) \leq {\hom}(K_{d+1},H_{\rm WR})^\frac{n}{d+1} $$ for all
$n$-vertex, $d$-regular $G$. This verifies a conjecture of Galvin.
