A Generalization of Clique Polynomials and Graph Homomorphism

The clique polynomial of a graph g is the ordinary generating function of the number of complete subgraphs (cliques) of g. in this paper, we introduce a new vertex-weighted version of these polynomials. we also show that these weighted clique polynomials have always a real root provided that the weights are non-negative real numbers. as an application, we obtain a no-homomorphism criteria based on the largest real root of our vertex-weighted clique polynomial.