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 of B and of its induced subgraphs B[X] and B[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).