Let G be a simple connected graph of order at least 2. An i-subset of  is a subset of  of cardinality i. An i-clique is an i-subset which induces a complete subgraph of G. The clique common neighborhood polynomial of G is given by   where  is the number of i-cliques in G with common neighborhood cardinality equal to j and  is the cardinality of a maximum clique in G, called the clique number of G. In this paper, we established the clique common neighborhood polynomials of the special graphs such as the complete graph, complete bipartite graph and complete q-partite graph. Moreover, we have shown that the clique polynomial is a special evaluation of the clique common neighborhood polynomial at

clique, clique polynomial, clique common neighborhood polynomial