On graphs with distance Laplacian eigenvalues of multiplicity $n-4$

TL;DR

This paper characterizes connected graphs with specific multiplicities of the distance Laplacian eigenvalues, especially focusing on the spectral radius and the eigenvalue equal to the number of vertices.

Contribution

It extends previous work by characterizing graphs with the distance Laplacian spectral radius of multiplicity n-4 and related eigenvalue multiplicities.

Findings

Graphs with distance Laplacian spectral radius of multiplicity n-4 are characterized.

Graphs where the eigenvalue n has multiplicity n-4 are fully determined.

The study generalizes earlier characterizations for eigenvalue multiplicities n-3.

Abstract

Let $G$ be a connected simple graph with $n$ vertices. The distance Laplacian matrix $D^{L}(G)$ is defined as $D^L(G)=Diag(Tr)-D(G)$, where $Diag(Tr)$ is the diagonal matrix of vertex transmissions and $D(G)$ is the distance matrix of $G$. The eigenvalues of $D^{L}(G)$ are the distance Laplacian eigenvalues of $G$ and are denoted by $\partial_{1}^{L}(G)\geq \partial_{2}^{L}(G)\geq \dots \geq \partial_{n}^{L}(G)$. The largest eigenvalue $\partial_{1}^{L}(G)$ is called the distance Laplacian spectral radius. Lu et al. (2017), Fernandes et al. (2018) and Ma et al. (2018) completely characterized the graphs having some distance Laplacian eigenvalue of multiplicity $n-3$. In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity $n-4$ together with one of the distance Laplacian eigenvalue as $n$ of multiplicity either 3 or 2. Further, we completely determine the graphs for which the distance Laplacian eigenvalue $n$ is of multiplicity $n-4$.