Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity n − 3

Abstract

Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on a related problem: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a graph is considered here. Let ρn−1(G) and ν(G) be the second smallest normalized Laplacian eigenvalue and the independence number of a graph G, respectively. As the main conclusions, two families of n-vertex connected graphs with some normalized Laplacian eigenvalue of multiplicity n−3 are determined: graphs with ρn−1(G)=1 and graphs with ρn−1(G)≠1 and ν(G)=3. Moreover, it is proved that these graphs are determined by their normalized Laplacian spectrum.