The full chracterazation of the graphs with a L-eigenvalue of multiplicity $n-3$

TL;DR

This paper fully characterizes connected graphs with a Laplacian eigenvalue of multiplicity n-3, completing the classification for graphs with three distinct Laplacian eigenvalues in this category.

Contribution

It determines the structure of graphs in g(n,n-3) with three distinct Laplacian eigenvalues, completing the previous partial classifications.

Findings

Graphs with n-3 multiplicity Laplacian eigenvalue are fully characterized.

The structure of graphs with three distinct Laplacian eigenvalues in this class is identified.

The classification extends previous results for specific eigenvalue multiplicities.

Abstract

Let $\mathcal{G}(n,k)$ be the set of connected graphs of order $n$ with one of the Laplacian eigenvalue having multiplicity $k$. It is well known that $\mathcal{G}(n,n-1)=\{K_n\}$. The graphs of $\mathcal{G}(n,n-2)$ are determined by Das, and the graphs of $\mathcal{G}(n,n-3)$ with four distinct Laplacian eigenvalues are determined by Mohammadian et al. In this paper, we determine the graphs of $\mathcal{G}(n,n-3)$ with three distinct Laplacian eigenvalues, and then the full characterization of the graphs in $\mathcal{G}(n,n-3)$ is completed.