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

TL;DR

This paper characterizes specific families of graphs with a normalized Laplacian eigenvalue of multiplicity n-3, identifying their structure and spectral uniqueness, thus advancing understanding of eigenvalue multiplicities in graph spectra.

Contribution

It identifies and characterizes two families of connected graphs with a normalized Laplacian eigenvalue of multiplicity n-3 and proves their spectral determination.

Findings

Identified graphs with normalized Laplacian eigenvalue of multiplicity n-3.

Characterized graphs with eigenvalue -1 and those with independence number not equal to 2.

Proved these graphs are uniquely determined by their spectrum.

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 $\rho_{n-1}(G)$ and $\nu(G)$ 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 $\rho_{n-1}(G)=-1$ and graphs with $\rho_{n-1}(G)\neq -1$ and $\nu(G)\neq 2$. Moreover, it is proved that these graphs are determined by their spectrum.