On the eccentric distance sum of graphs

The eccentric distance sum is a novel topological index that offers a vast potential for structure activity/property relationships. For a graph G, it is defined as ξ d ( G ) = ∑ v ∈ V ε ( v ) D ( v ) , where ε ( v ) is the eccentricity of the vertex v and D ( v ) = ∑ u ∈ V ( G ) d ( u , v ) is the sum of all distances from the vertex v. Motivated by [G. Yu, L. Feng, A. Ilić, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 934–944], in this paper we characterize the extremal trees and graphs with maximal eccentric distance sum. Various lower and upper bounds for the eccentric distance sum in terms of other graph invariants including the Wiener index, the degree distance, eccentric connectivity index, independence number, connectivity, matching number, chromatic number and clique number are established. In addition, we present explicit formulae for the values of eccentric distance sum for the Cartesian product, applied to some graphs of chemical interest (like nanotubes and nanotori).