POLYNOMIAL REPRESENTATIONS OF THE BICLIQUE NEIGHBORHOOD OF GRAPHS

An induced subgraph H of a graph G is a balanced biclique of G if H congruent to K-i,K-i for some i is an element of{1, 2, ... , left perpendicular vertical bar V(G)vertical bar/2 right perpendicular}. The balanced biclique polynomial of G is given by b(G, x) = Sigma(beta(G)/2)(i=1) b(i)(G)x(2i), where b(i)(G) is the number of balanced bicliques of G of order 2i and beta(G) is the cardinality of a maximum balanced biclique of G. The balanced biclique neighborhood polynomial of G which counts the balanced bicliques of G with corresponding neighborhood system cardinality is given by bn(G; x, y) = Sigma(n-2)(j=0) Sigma(beta(G)/2)(i=1) b(ij)(G)x(2i)y(j), where b(ij)(G) is the number of balanced bicliques of G of order 2i with neighborhood cardinality equal to j and beta(G) is the cardinality of a maximum balanced biclique of G. In this paper, we establish the balanced biclique neighborhood polynomial of some special graphs such as cycle, complete graph, complete bipartite graph, and complete q-partite graph.