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

The Hebrew University of Jerusalem

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 - 1 Sep 2023

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

10.1007/s11856-023-2519-3

Cite this

@article{4d819bba0791465eb4931f521ae54c67,

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 = "Publisher Copyright: {\textcopyright} 2023, The Hebrew University of Jerusalem.",

year = "2023",

month = sep,

day = "1",

doi = "10.1007/s11856-023-2519-3",

language = "American English",

volume = "256",

pages = "479--511",

journal = "Israel Journal of Mathematics",

issn = "0021-2172",

publisher = "Springer New York",

number = "2",

}

TY - JOUR

T1 - On the d-dimensional algebraic connectivity of graphs

AU - Lew, Alan

AU - Nevo, Eran

AU - Peled, Yuval

AU - Raz, Orit E.

PY - 2023/9/1

Y1 - 2023/9/1

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=85173760337&partnerID=8YFLogxK

U2 - 10.1007/s11856-023-2519-3

DO - 10.1007/s11856-023-2519-3

M3 - Article

SN - 0021-2172

VL - 256

SP - 479

EP - 511

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

IS - 2

ER -

On the d-dimensional algebraic connectivity of graphs

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this