Graph operations and upper bounds on graph homomorphism counts

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), where 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 holds for all n‐vertex, d‐regular G. In particular, we show that if H_WR is the complete looped path on three vertices, also known as the Widom–Rowlinson graph, then for all n‐vertex, d‐regular G. This verifies a conjecture of Galvin.