Let hom(H,G) denote the number of homomorphisms from a graph H to a
graph G. Sidorenko's conjecture asserts that for any bipartite graph H, and
a graph G we have hom(H,G)≥v(G)v(H)(v(G)2hom(K2,G))e(H), where v(H),v(G)
and e(H),e(G) denote the number of vertices and edges of the graph H and
G, respectively. In this paper we prove Sidorenko's conjecture for certain
special graphs G: for the complete graph Kq on q vertices, for a K2
with a loop added at one of the end vertices, and for a path on 3 vertices
with a loop added at each vertex. These cases correspond to counting colorings,
independent sets and Widom-Rowlinson colorings of a graph H. For instance,
for a bipartite graph H the number of q-colorings ch(H,q)
satisfies ch(H,q)≥qv(H)(qq−1)e(H).
In fact, we will prove that in the last two cases (independent sets and
Widom-Rowlinson colorings) the graph H does not need to be bipartite. In all
cases, we first prove a certain correlation inequality which implies
Sidorenko's conjecture in a stronger form.Comment: Two references added and Remark 2.1 is expande