On generalized distance spectral radius and generalized distance energy of graphs

Abstract

For a simple connected graph [Formula: see text], let [Formula: see text] and [Formula: see text] be the distance matrix and the diagonal matrix of the vertex transmissions, respectively. The convex linear combination [Formula: see text] of [Formula: see text] and [Formula: see text] is defined as, [Formula: see text], [Formula: see text]. The matrix [Formula: see text], known as generalized distance matrix, is effective in merging the distance spectral and distance signless Laplacian spectral theories. In this paper, we study the spectral radius and energy of the generalized distance matrix [Formula: see text] of a graph [Formula: see text]. We obtain bounds for the generalized distance spectral radius and generalized distance energy of connected graphs in terms of various parameters associated with the structure of graph.