If G is a graph, its clique graph K(G) is the intersection graph of all its (maximal) cliques. The complex G<FONT FACE=Symbol></FONT> of a graph G is the simplicial complex whose simplexes are the vertex sets of the complete subgraphs of G. Here we study a sufficient condition for G<FONT FACE=Symbol></FONT> and K(G)<FONT FACE=Symbol></FONT> to be homotopic. Applying this result to Whitney triangulations of surfaces, we construct an infinite family of examples which solve in the affirmative Prisner's open problem 1 in Graph Dynamics (Longman, Harlow, 1995): Are there finite connected graphs G that are periodic under K and where the second modulo 2 Betti number is greater than 0?
clique graphs; clique convergence; Whitney triangulations; clean triangulations; simplicial complexes; modulo 2 Betti numbers
