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.

Jordán 和 Tanigawa 最近引入了图 G 的 d 维代数连通性 \（a_d（G）\）。这是 G 的 d 维刚度的定量测量，它推广了经过充分研究的图谱扩展概念。我们提出了一个新的 \（a_d（G）\） 下限，该下限是根据 G 的某些子图的光谱扩展来定义的，这些子图与将其顶点划分为 d 部分相关联。特别是，我们获得了图 G 刚度的新充分条件。作为第一个应用，我们证明了每个 \（d\ge 2\） 和 \（k\ge 2d+1\） 存在一个无限的 k 规则 d 刚性扩展器图族。从推测上，不存在这样的 2d 正则图族。其次，我们展示了 \（a_d（K_n）\ge \frac{1}{2}\left\lfloor \frac{n}{d}\right\rfloor \），我们推测它本质上是紧密的。此外，我们还研究了如果 G 是一个最小 d 刚性图，则 \（a_d（G）\） 会达到的极值。