# On graphs with $m(\partial^L_1)=n-3$

## TL;DR

This paper characterizes connected graphs with a specific spectral property, namely those whose largest distance Laplacian eigenvalue has multiplicity exactly n-3, extending previous classifications for multiplicities n-1 and n-2.

## Contribution

It provides a complete characterization of graphs where the largest distance Laplacian eigenvalue has multiplicity n-3, filling a gap in spectral graph theory.

## Key findings

- Graphs with $m(oundary^L_1)=n-3$ are fully characterized.
- Extends previous results for multiplicities n-1 and n-2.
- Contributes to understanding the spectral properties of graphs.

## Abstract

Let $\partial^L_1\ge\partial^L_2\ge\cdots\ge\partial^L_n$ be the distance Laplacian eigenvalues of a connected graph $G$ and $m(\partial^L_i)$ the multiplicity of $\partial^L_i$. It is well known that the graphs with $m(\partial^L_1)=n-1$ are complete graphs. Recently, the graphs with $m(\partial^L_1)=n-2$ have been characterized by Celso et al. In this paper, we completely determine the graphs with $m(\partial^L_1)=n-3$.