Convergence of iterated clique graphs

Abstract C ( G ) denotes the clique graph of G and C n ( G ):= C ( C n −1( G )) is the n th iterated clique graph. A graph G clique-converges to a set M ={ F , C ( F ), C 2 ( F ),…, C p −1( F } of graphs if C p ( F )= F and C m ( G )= F for some integer m . The simplicial complex G ↑ of a graph G has the vertex sets of all complete subgraphs of G as simplices. By β i we denote the i th modulo 2 Betti number of a complex. A vertex d of G is dominated by a neighbour z if every other neighbour of d is also adjacent to z . The completely pared graph P ∞ ( G ) of a graph G is the graph which we finally obtained by deleting successively dominated vertices. the main results of this paper are the following: (1) If G clique-converges to { F , C ( F ),…}, then β 1 ( G ↑ )= β 1 ( F ↑ ) and β 2 ( G ↑ )⩾ β 2 ( F ↑ )= β 2 ( C ( F ) ↑ )=⋯. (2) If P ∞ ( G ) is triangleless, then G clique-converges to { P ∞ ( G )} or to { P ∞ ( G ), CP ∞ ( G )}.