On weighted graph homomorphisms

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.

Matchings and independent sets of a fixed size in regular graphs

“…From (10) we easily obtain the claimed bound, following the steps of the derivation of the second bound in (1) from (8). We prove (10) by using a more general result on graph homomorphisms.…”

“…We begin with the second bound in (1). We use a result from [8], which states that for any λ > 0 and any d-regular N-vertex bipartite graph G, the weighted independent set partition function satisfies…”

“…If G is bipartite with bipartition classes E and O and ≺ satisfies u ≺ v for all u ∈ E, v ∈ O then Theorem 3.1 reduces to the main result of [8].…”

“…Let C be an integer such that C λ is also an integer, and let H C be the graph which consists of an independent set of size C λ and a complete looped graph on C vertices, with a complete bipartite graph joining the two. As described in [8] we have, for any graph G on N vertices,…”

Sandwich bounds for joint entropy

“…Shearer's lemma was (implicitly) introduced in [4], and Kahn [7] stated an extension using the more familiar entropy notation. Recent applications of Shearer's lemma to difficult problems in combinatorics include [8], [7], [9], and [10]. Radhakrishnan [5] provides a nice survey of entropy ideas used for counting and various applications.…”