The fundamental group of the clique graph

Given a finite connected bipartite graph B = ( X , Y ) we consider the simplicial complexes of complete subgraphs of the square B 2 of B and of its induced subgraphs B 2 [ X ] and B 2 [ Y ] . We prove that these three complexes have isomorphic fundamental groups. Among other applications, we conclude that the fundamental group of the complex of complete subgraphs of a graph G is isomorphic to that of the clique graph K ( G ) , the line graph L ( G ) and the total graph T ( G ) .