The distribution of eigenvalues of graphs
type: Item
instance of: scholarly article
title: The distribution of eigenvalues of graphs
zbMATH Open document ID: 0820.05047
DOI: 10.1016/0024-3795(93)00135-M
author: Dasong Cao
author: Hong Yuan
published in: Linear Algebra and its Applications
publication date: 1995-08-27T00:00:00Z
review text: Let \(\lambda_ \kappa (G)\) be the \(\kappa\)-th largest eigenvalue of \(G\). A real number \(\gamma\) is said to be the limit point of the \(\kappa\)-th largest eigenvalues of graphs if there exists a sequence \(\{G_ n\}\) of graphs such that \(\lim_{n \to \infty} \lambda_ k (G_ n) = \gamma\). This paper finds the smallest limit point of the \(\kappa\)-th largest eigenvalues and an upper bound of the limit points of the \(\kappa\)-th smallest eigenvalues and proves that for \(\kappa \geq 4\) there exists a gap beyond the smallest limit point in which no point is the limit point of the \(\kappa\)-th largest eigenvalues. Finally, this paper shows that (1) \(\lambda_ 3 (G) < - 1\) iff \(G \cong P_ 3\); (2) \(\lambda_ 3 (G) = - 1\) iff \(G^ c\) is isomorphic to a complete bipartite graph plus isolated vertices; (3) there exist no graphs such that \(-1 < \lambda_ 3 (G) < (1 - \sqrt 5)/2\).
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 753892
zbMATH Keywords: eigenvalue
zbMATH Keywords: limit point
zbMATH Keywords: gap
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 753892