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

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(T r) − D(G), where Diag(T r) 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 byis 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.