Full characterization of graphs having certain normalized Laplacian eigenvalue of multiplicity $n-3$

Abstract

Let $G$ be a connected simple graph of order $n$. Let ${\rho }_1(G)\ge {\rho }_2(G)\ge \cdots \ge {\rho }_{n-1}(G)>{\rho }_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m({\rho }_i)$ the multiplicity of the normalized Laplacian eigenvalue ${\rho }_i$. Let $\nu (G)$ be the independence number of $G$. In this paper, we give a full characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, which answers a remaining problem in [S. Sun, K.C. Das, On the multiplicities of normalized Laplacian eigenvalues of graphs, Linear Algebra Appl. 609 (2021) 365-385], $i.e.,$ there is no graph with $m({\rho }_1)=n-3$ ($n\ge 6$) and $\nu (G)=2$. Moreover, we confirm that all the graphs with $m({\rho }_1)=n-3$ are determined by their normalized Laplacian spectra.

References in corpus (1)

• On graphs with some normalized Laplacian eigenvalue of extremal multiplicity