On the d-dimensional algebraic connectivity of graphs

Alan Lew; Eran Nevo; Yuval Peled; Orit E. Raz

doi:10.1007/s11856-023-2519-3

On the d-dimensional algebraic connectivity of graphs

Alan Lew^*, Eran Nevo, Yuval Peled, Orit E. Raz

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

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, ⌈n2d⌉−2d+1≤ad(Kn)≤2n3(d−1)+13.

Original language	American English
Pages (from-to)	479-511
Number of pages	33
Journal	Israel Journal of Mathematics
Volume	256
Issue number	2
DOIs	https://doi.org/10.1007/s11856-023-2519-3
State	Published - Sep 2023

Bibliographical note

Funding Information:
Alan Lew and Eran Nevo were partially supported by the Israel Science Foundation grant ISF-2480/20. Acknowledgments

Funding Information:
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.

Publisher Copyright:
© 2023, The Hebrew University of Jerusalem.

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title = "On the d-dimensional algebraic connectivity of graphs",

abstract = "The d-dimensional algebraic connectivity ad(G) of a graph G = (V,E), introduced by Jord{\'a}n 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.",

author = "Alan Lew and Eran Nevo and Yuval Peled and Raz, {Orit E.}",

note = "Funding Information: Alan Lew and Eran Nevo were partially supported by the Israel Science Foundation grant ISF-2480/20. Acknowledgments Funding Information: 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. Publisher Copyright: {\textcopyright} 2023, The Hebrew University of Jerusalem.",

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AU - Peled, Yuval

AU - Raz, Orit E.

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AB - 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.

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On the d-dimensional algebraic connectivity of graphs

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