On weighted graph homomorphisms

TL;DR

This paper proves that for regular bipartite graphs, the maximum number of homomorphisms to any graph occurs when the graph is a disjoint union of complete bipartite graphs, extending previous combinatorial results.

Contribution

It generalizes Kahn's result to weighted graph homomorphisms and provides asymptotic formulas based on a simple parameter of the target graph.

Findings

Maximum homomorphisms occur at disjoint unions of K_{n,n}

Asymptotic formulas for the logarithm of homomorphism counts

Weighted versions relate to physical models with constraints

Abstract

For given graphs $G$ and $H$, let $|Hom(G,H)|$ denote the set of graph homomorphisms from $G$ to $H$. We show that for any finite, $n$-regular, bipartite graph $G$ and any finite graph $H$ (perhaps with loops), $|Hom(G,H)|$ is maximum when $G$ is a disjoint union of $K_{n,n}$'s. This generalizes a result of J. Kahn on the number of independent sets in a regular bipartite graph. We also give the asymptotics of the logarithm of $|Hom(G,H)|$ in terms of a simply expressed parameter of $H$. We also consider weighted versions of these results which may be viewed as statements about the partition functions of certain models of physical systems with hard constraints.