CLIQUE COMMON NEIGHBORHOOD POLYNOMIAL OF GRAPHS

Let G be a simple connected graph of order at least 2. An i-subset of V(G) is a subset of V(G) 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 ccn(G; x, y) = Sigma(j=0) (n-i omega(G)) Sigma(i=1) cij(G)x(i) y(j), where c(ij)(G) is the number of i-cliques in G with common neighborhood cardinality equal to j and omega(G) 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 y = 1.