文献类型：期刊文献

英文题名：Spectral radius conditions for the rigidity of graphs

作者：Fan, Dandan[1,2];Huang, Xueyi[1];Lin, Huiqiu[1]

机构：[1]East China Univ Sci & Technol, Sch Math, Shanghai 200237, Peoples R China;[2]Xinjiang Agr Univ, Coll Math & Phys, Urumqi 830052, Xinjiang, Peoples R China

年份：2023

卷号：30

期号：2

外文期刊名：ELECTRONIC JOURNAL OF COMBINATORICS

收录：;WOS:【SCI-EXPANDED(收录号:WOS:000975535100001)】；

基金：Acknowledgments ? Sponsored by Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant Nos. 2022D01B103) . ? Corresponding author. Supported by the National Natural Science Foundation of China (Grant Nos. 12271162) , Natural Science Foundation of Shanghai (No. 22ZR1416300) .

语种：英文

摘要：Rigidity is the property of a structure that does not flex under an applied force. In the past several decades, the rigidity of graphs has been widely studied in discrete geometry and combinatorics. Laman (1970) obtained a combinatorial characteriza-tion of rigid graphs in R2. Lov ' asz and Yemini (1982) proved that every 6-connected graph is rigid in R2. Jackson and Jord ' an (2005) strengthened this result, and showed that every 6-connected graph is globally rigid in R2. Thus every graph with alge-braic connectivity greater than 5 is globally rigid in R2. In 2021, Cioaba, Dewar and Gu improved this bound, and proved that every graph with minimum degree at least 6 and algebraic connectivity greater than 2+ 1 delta-1 (resp., 2+ delta 2-1) is rigid (resp., globally rigid) in R2. In this paper, we study the rigidity of graphs in R2 from the viewpoint of adjacency eigenvalues. Specifically, we provide a spectral radius con-dition for the rigidity (resp., globally rigidity) of 2-connected (resp., 3-connected) graphs with given minimum degree. Furthermore, we determine the unique graph attaining the maximum spectral radius among all minimally rigid graphs of order n.