A family of multiply extended grids

We construct an infinite family{_n}_n=5 of finite connected graphs_n that are multiple extensions of the well-known extended grid discovered in [1] (which is isomorphic to₅). The graphs_n are locally_n–1 forn > 5, and have the following property: the automorphism groupG(n) of_n permutes transitively the maximal cliques of_n (which aren-cliques) and the stabilizer of somen-clique of_n inG(n) induces_n on the vertices of. Furthermore we show that the clique complexes of the graphs_n are simply connected.