Rigidity Expander Graphs

Jordán and Tanigawa recently introduced the d-dimensional algebraic connectivity ad(G) of a graph G. This is a quantitative measure of the d-dimensional rigidity of G which generalizes the well-studied notion of spectral expansion of graphs. We present a new lower bound for ad(G) defined in terms of the spectral expansion of certain subgraphs of G associated with a partition of its vertices into d parts. In particular, we obtain a new sufficient condition for the rigidity of a graph G. As a first application, we prove the existence of an infinite family of k-regular d-rigidity-expander graphs for every d≥2 and k≥2d+1. Conjecturally, no such family of 2d-regular graphs exists. Second, we show that ad(Kn)≥12nd, which we conjecture to be essentially tight. In addition, we study the extremal values ad(G) attains if G is a minimally d-rigid graph.