  The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$,
introduced by Jord\'an 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 $\mathbb{R}^d$.
  Here, we analyze the $d$-dimensional algebraic connectivity of complete
graphs. In particular, we show that, for $d\geq 3$, $a_d(K_{d+1})=1$, and for
$n\geq 2d$, \[ \left\lceil\frac{n}{2d}\right\rceil-2d+1\leq a_d(K_n) \leq
\frac{2n}{3(d-1)}+\frac{1}{3}. \]
