The distribution of eigenvalues of graphs
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 \(-1 < \lambda_ 3 (G) < (1 - \sqrt 5)/2\).
scientific article; zbMATH DE number 753892