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

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 $n-4$.
Comment: 11 pages