{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T11:06:10Z","timestamp":1770980770303,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 characterization of rigid graphs in $\\mathbb{R}^2$. Lov\u00e1sz and Yemini (1982) proved that every $6$-connected graph is rigid in $\\mathbb{R}^2$. Jackson and Jord\u00e1n (2005) strengthened this result, and showed that every $6$-connected graph is globally rigid in $\\mathbb{R}^2$. Thus every graph with algebraic connectivity greater than $5$ is globally rigid in $\\mathbb{R}^2$. In 2021, Cioab\u0103, Dewar and Gu improved this bound, and proved that every graph with minimum degree at least $6$ and algebraic connectivity greater than $2+\\frac{1}{\\delta-1}$ (resp., $2+\\frac{2}{\\delta-1}$) is rigid (resp., globally rigid) in $\\mathbb{R}^2$. In this paper, we study the rigidity of graphs in $\\mathbb{R}^2$ from the viewpoint of adjacency eigenvalues. Specifically, we provide a spectral radius condition 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 $n$.<\/jats:p>","DOI":"10.37236\/11308","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T11:34:33Z","timestamp":1682076873000},"source":"Crossref","is-referenced-by-count":2,"title":["Spectral Radius Conditions for the Rigidity of Graphs"],"prefix":"10.37236","volume":"30","author":[{"given":"Dandan","family":"Fan","sequence":"first","affiliation":[]},{"given":"Xueyi","family":"Huang","sequence":"additional","affiliation":[]},{"given":"Huiqiu","family":"Lin","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,4,21]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p14\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T11:34:33Z","timestamp":1682076873000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i2p14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,21]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,4,7]]}},"URL":"https:\/\/doi.org\/10.37236\/11308","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,4,21]]},"article-number":"P2.14"}}