Entropy, Graph Homomorphisms, and Dissociation Sets

Given two graphs G and H, the mapping of [Formula: see text] is called a graph homomorphism from G to H if it maps the adjacent vertices of G to the adjacent vertices of H. For the graph G, a subset of vertices is called a dissociation set of G if it induces a subgraph of G containing no paths of order three, i.e., a subgraph of a maximum degree, which is at most one. Graph homomorphisms and dissociation sets are two generalizations of the concept of independent sets. In this paper, by utilizing an entropy approach, we provide upper bounds on the number of graph homomorphisms from the bipartite graph G to the graph H and the number of dissociation sets in a bipartite graph G.