The d-dimensional algebraic connectivity a_d(G) of a graph G = (V,E), introduced by Jordán and Tanigawa, is a quantitative measure of the d-dimensional rigidity of G that is defined in terms of the eigenvalues of stiffness matrices (which are analogues of the graph Laplacian) associated to mappings of the vertex set V into ℝ^d.

Here, we analyze the d-dimensional algebraic connectivity of complete graphs. In particular, we show that, for d ≥ 3, a_d(K_d+1) = 1, and for n ≥ 2d,

$$\left\lceil {{n \over {2d}}} \right\rceil - 2d + 1 \le {a_d}({K_n}) \le {{2n} \over {3(d - 1)}} + {1 \over 3}.$$

中文翻译：

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关于图的 d 维代数连通性

图G = ( V,E ) 的d维代数连通性a _d ( G ) 由 Jordán 和 Tanikawa 引入，是G的d维刚度的定量度量，它是根据刚度特征值定义的与顶点集V到 ℝ ^d的映射相关的矩阵（与拉普拉斯图类似）。

在这里，我们分析完全图的d维代数连通性。特别是，我们证明，对于d ≥ 3，a _d ( K _{d +1} ) = 1，并且对于n ≥ 2 d，

$$\left\lceil {{n \over {2d}}} \right\rceil - 2d + 1 \le {a_d}({K_n}) \le {{2n} \over {3(d - 1)} } + {1 \ 超过 3}.$$