On self-clique graphs / Ong Poh Hwa - Ong, Poh Hwa (2010)
The clique graph of a graph G is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques have non-empty intersection. A graph G is self-clique if it is isomorphic to its clique graph. Clique graphs have been studied for some time. However, not much is known about self-clique graphs. Self-clique graphs were ¯rst introduced and studied by Escalante [Abh. Math. Sem. Univ. Hamburg 39 (1973) 59-68]. Since then, self-clique graphs have been characterized for some classes of graphs. Chia [Discrete Math. 212 (2000) 185-189] gave a characterization of connected self-clique graphs in which all cliques have size two, except for precisely one clique. The main objective in this thesis is to characterize all connected self-clique graphs with given clique sizes. Some known results on the characterizations of clique graphs and self-clique graphs are presented. We obtain a characterization for the set of all connected self-clique graphs having all cliques but two of size 2. We also give several results on connected self-clique graphs in which each clique has the same size k for k = 2 and k = 3.