On bicliques and the second clique graph of suspensions

Abstract

The clique graphK(G) of a graph G is the intersection graph of the set of all (maximal) cliques of G. The second clique graphK2(G)ofG is defined as K2(G)=K(K(G)). The main motivation for this work is to attempt to characterize the graphs G that maximize |K2(G)|, as has been done for |K(G)| by Moon and Moser in 1965.The suspensionS(G) of a graph G is the graph that results from adding two non-adjacent vertices to the graph G, that are adjacent to every vertex of G. Using a new biclique operator B that transforms a graph G into its biclique graphB(G), we found the characterization K2(S(G))≅B(K(G)). We also found a characterization of the graphs G, that maximize |B(G)|.Here, a biclique (X,Y) of G is an ordered pair of subsets of vertices of G (not necessarily disjoint), such that every vertex x∈X is adjacent or equal to every vertex y∈Y, and such that (X,Y) is maximal under component-wise inclusion. The biclique graph B(G) of the graph G, is the graph whose vertices are the bicliques of G and two vertices (X,Y) and (X′,Y′) are adjacent, if and only if X∩X′≠0̸ or Y∩Y′≠0̸.