Let H be a graph allowing loops as well as vertex and edge weights. We
prove that, for every triangle-free graph G without isolated vertices, the
weighted number of graph homomorphisms hom(G,H) satisfies the inequality hom(G,H)≤uv∈E(G)∏hom(Kdu,dv,H)1/(dudv),
where du denotes the degree of vertex u in G. In particular, one has hom(G,H)1/∣E(G)∣≤hom(Kd,d,H)1/d2 for every d-regular
triangle-free G. The triangle-free hypothesis on G is best possible. More
generally, we prove a graphical Brascamp-Lieb type inequality, where every edge
of G is assigned some two-variable function. These inequalities imply tight
upper bounds on the partition function of various statistical models such as
the Ising and Potts models, which includes independent sets and graph
colorings.
For graph colorings, corresponding to H=Kq, we show that the
triangle-free hypothesis on G may be dropped; this is also valid if some of
the vertices of Kq are looped. A corollary is that among d-regular graphs,
G=Kd,d maximizes the quantity cq(G)1/∣V(G)∣ for every q and d,
where cq(G) counts proper q-colorings of G.
Finally, we show that if the edge-weight matrix of H is positive
semidefinite, then hom(G,H)≤v∈V(G)∏hom(Kdv+1,H)1/(dv+1). This implies that among d-regular graphs, G=Kd+1
maximizes hom(G,H)1/∣V(G)∣. For 2-spin Ising models, our results give a
complete characterization of extremal graphs: complete bipartite graphs
maximize the partition function of 2-spin antiferromagnetic models and cliques
maximize the partition function of ferromagnetic models.
These results settle a number of conjectures by Galvin-Tetali, Galvin, and
Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture
by Kahn.Comment: 30 page