Relation between generalized distance spread and some topological indices

Let [Formula: see text] be a connected graph. The generalized distance matrix is defined as [Formula: see text], where [Formula: see text], [Formula: see text] is the distance matrix whose [Formula: see text]th entry is the distance between the [Formula: see text]th and [Formula: see text]th vertices in [Formula: see text] and [Formula: see text] is the diagonal matrix where each nonzero entry corresponds to the total distance to a vertex. The generalized distance spread, [Formula: see text] was defined as the difference between the largest eigenvalue and the smallest eigenvalue of [Formula: see text]. In this paper some lower bounds for [Formula: see text] in terms of [Formula: see text], the Wiener index and [Formula: see text] is presented and the equality cases are discussed. For a graph with prescribed diameter, a lower bound for [Formula: see text] as function of the vertex transmissions, distances between vertices, the diameter and [Formula: see text] is presented. A lower bound for this graph invariant in the class of trees is given. Additionally, lower bounds for [Formula: see text] as function of some known topological indices are presented. Some closed formulas for [Formula: see text] of known vertex-transitive graphs such as the polyhex nanotorus and hypercube are presented as applications of previous results. Moreover, some lower bounds (as function of [Formula: see text]) for some known molecular graphs are also presented.