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For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of$G$ to the vertices of $H$ that preserves edge adjacency. We consider thefollowing extremal enumerative question: for a given $H$, which connected$n$-vertex graph with minimum degree $\delta$ maximizes the number of$H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, thecomplete bipartite graph $K_{\delta,n-\delta}$ is the unique maximizer. As acorollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$is the unique $k$-connected graph that maximizes the number of $H$-coloringsamong all $k$-connected graphs. Finally, we show that this conclusion does nothold for all regular $H$ by exhibiting a connected $n$-vertex graph withminimum degree $\delta$ which has more $K_{q}$-colorings (for sufficientlylarge $q$ and $n$) than $K_{\delta,n-\delta}$.

Maximizing $H$-colorings of connected graphs with fixed minimum degree