Entropy, Graph Homomorphisms, and Dissociation Sets.

Given two graphs and , the mapping of f:V(G)→V(H) is called a graph homomorphism from to if it maps the adjacent vertices of to the adjacent vertices of . For the graph , a subset of vertices is called a dissociation set of if it induces a subgraph of 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 to the graph and the number of dissociation sets in a bipartite graph .