Eigenvalue localization and Geršgorin disc-related problems on distance and distance-related matrices of graphs

Abstract

The distance, distance signless Laplacian and distance Laplacian matrix of a simple connected graph [Formula: see text] are denoted by [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] is the diagonal matrix of vertex transmission. Ger[Formula: see text]gorin discs for any [Formula: see text] square matrix [Formula: see text] are the discs [Formula: see text], where [Formula: see text]. The famous Ger[Formula: see text]gorin disc theorem says that all the eigenvalues of a square matrix lie in the union of the Ger[Formula: see text]gorin discs of that matrix. In this paper, some classes of graphs are studied for which the smallest Ger[Formula: see text]gorin disc contains every distance and distance signless Laplacian eigenvalues except the spectral radius of the corresponding matrix. For all connected graphs, a lower bound and for trees, an upper bound of every distance signless Laplacian eigenvalues except the spectral radius is given in this paper. These bounds are comparatively better than the existing bounds. By applying these bounds, we find some infinite classes of graphs for which the smallest Ger[Formula: see text]gorin disc contains every distance signless Laplacian eigenvalues except the spectral radius of the distance signless Laplacian matrix. For the distance Laplacian eigenvalues, we have given an upper bound and then find a condition for which the smallest Ger[Formula: see text]gorin disc contains every distance Laplacian eigenvalue of the distance Laplacian matrix. These results give partial answers from some questions that are raised in [2].