On bicliques and the second clique graph of suspensions

The clique graph K(G) of a graph G is the intersection graph of the set of all (maximal) cliques of G. The second clique graph K-2(G) of G is defined as K-2(G) = K(K(G)). The main motivation for this work is to attempt to characterize the graphs G that maximize vertical bar K-2(G)vertical bar, as has been done for vertical bar K(G)vertical bar by Moon and Moser in 1965. The suspension S(G) of a graph G is the graph that results from adding two nonadjacent 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 graph B(G), we found the characterization K-2(S(G)) congruent to B(K(G)). We also found a characterization of the graphs G, that maximize vertical bar B(G)vertical bar. 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 is an element of 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 boolean AND X' not equal empty set or Y n Y' not equal empty set . (c) 2019 Elsevier B.V. All rights reserved.