On Self-Clique Graphs all of whose Cliques have Equal Size

Abstract

The clique graph of a graph $G$ is the graph whose vertex set is the set of cliques of $G$ and two vertices are adjacent if and only if the corresponding cliques have non-empty intersection. A graph is self-clique if it is isomorphic to its clique graph. In this paper, we present several results on connected self-clique graphs in which each clique has the same size $k$ for $k=2$ and $k=3$.