The d-dimensional algebraic connectivity ad(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, ad(Kd+1) = 1, and for n ≥ 2d, ⌈n2d⌉−2d+1≤ad(Kn)≤2n3(d−1)+13.
Bibliographical noteFunding Information:
Alan Lew and Eran Nevo were partially supported by the Israel Science Foundation grant ISF-2480/20. Acknowledgments
Part of this research was done while A.L. was a postdoctoral researcher at the Einstein Institute of Mathematics at the Hebrew University. We thank the anonymous referee for their helpful remarks and suggestions.
© 2023, The Hebrew University of Jerusalem.