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

2019/12/31 by Tian, Fenglei, Wong, Dein
#05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.13227

Abstract

Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a graph is considered here. Let ρ_n-1(G) and ν(G) be the second least 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 ρ_n-1(G)=-1 and graphs with ρ_n-1(G)≠ -1 and ν(G)≠ 2. Moreover, it is proved that these graphs are determined by their spectrum.

Related

• Full characterization of graphs having certain normalized Laplacian eigenvalue of multiplicity n-3
• On graphs with some normalized Laplacian eigenvalue of extremal multiplicity
• On graphs with three or four distinct normalized Laplacian eigenvalues
• The full chracterazation of the graphs with a L-eigenvalue of multiplicity n-3
• A few properties of the eigenvalues of normalized graph Laplacian