On cliques and bicliques

Abstract

Basic definitions are given in the next paragraph. We study second clique graphs of suspensions of graphs, K2(S(G)), and characterize them, in terms of an auxiliary biclique operator B which transforms a graph G into its biclique graph B(G). The characterization is then: K2(S(G))≅B(K(G)). We found a characterization of the graphs, G, that maximize |B(G)| for any given order n=|G|. This particular version of a biclique operator is new in the literature. The main motivation to study B(G) is an attempt to characterize the graphs G that maximize |K2(G)|, thus mimicking a result of Moon and Moser [12] that characterizes the graphs maximizing |K(G)|.The clique graph K(G) of a a graph G is the intersection graph of the set of all (maximal) cliques of G (and K2(G)=K(K(G))). The suspension S(G) of a graph G is the graph obtained from G by adding two new vertices which are adjacent to all other vertices, but not to each other. Here, a biclique (X, Y ) is an ordered pair of not necessarily disjoint subsets of vertices of G such that each x∈X is adjacent or equal to every y∈Y and such that (X, Y ) is maximal under component-wise inclusion. Finally B(G) is the graph whose vertices are the bicliques of G with adjacencies given by (X,Y)≃(X′,Y′) if and only if X∩X′≠∅ or Y∩Y′≠∅.