On graphs with distance Laplacian eigenvalues of multiplicity n−4

Abstract

AbstractLet G be a connected simple graph with n vertices. The distance Laplacian matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the diagonal matrix of vertex transmissions and [Formula: see text] is the distance matrix of G. The eigenvalues of [Formula: see text] are the distance Laplacian eigenvalues of G and are denoted by [Formula: see text]. The largest eigenvalue [Formula: see text] 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 [Formula: see text]. In this paper, we characterize the graphs having distance Laplacian spectral radius of multiplicity [Formula: see text] 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 [Formula: see text].

Keywords

• Distance matrix
• distance Laplacian matrix
• spectral radius
• multiplicity of distance Laplacian eigenvalue
• 05C50
• 05C12