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 ρ1(G)≥ρ2(G)≥⋯≥ρn−1(G)>ρn(G)=0 be the eigenvalues of the normalized Laplacian matrix L(G) of G. Denote by m(ρi) the multiplicity of the normalized Laplacian eigenvalue ρi. Let ν(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 Sun and Das (2021) [9], i.e., there is no graph with m(ρ1)=n−3 (n≥6) and ν(G)=2. Moreover, we confirm that all the graphs with m(ρ1)=n−3 are determined by their normalized Laplacian spectra.