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 by ∂ 1 L ( G ) ≥ ∂ 2 L ( G ) ≥ ⋯ ≥ ∂ n L ( G ) . The largest eigenvalue ∂ 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 eigenvalues 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 .