Rigidity Expander Graphs

Jordán and Tanigawa recently introduced the d-dimensional algebraic connectivity \(a_d(G)\) 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 \(a_d(G)\) 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 \(d\ge 2\) and \(k\ge 2d+1\). Conjecturally, no such family of 2d-regular graphs exists. Second, we show that \(a_d(K_n)\ge \frac{1}{2}\left\lfloor \frac{n}{d}\right\rfloor \), which we conjecture to be essentially tight. In addition, we study the extremal values \(a_d(G)\) attains if G is a minimally d-rigid graph.