On the eccentric distance sum of trees and unicyclic graphs

Abstract

Let G be a simple connected graph with the vertex set V (G). The eccentric distanceis the sum of all distances from the vertex v. In this paper we characterize the extremal unicyclic graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum. In addition, we characterize the extremal trees with given diameter and minimal eccentric distance sum.

The eccentric-distance sum of some graphs

“…Yu, Feng and Ilić [27] identified the extremal unicyclic graphs of given girth having the minimal and second minimal EDS; they also characterized trees with minimal EDS among the n-vertex trees of a given diameter. Hua, Xu and Shu [16] obtained the sharp lower bound on EDS of n-vertex cacti.…”

On the minimum eccentric distance sum of bipartite graphs with some given parameters

“…Yu et al [25] characterized the graphs among n-vertex unicyclic graphs with given girth having the minimal and second minimal eccentric distance sum, respectively. The extremal trees of given diameter having minimal eccentric distance sum were identified as well.…”

On the adjacent eccentric distance sum of graphs

“…Some mathematical properties of other molecular topological indices and their application for predicting biological and physical properties have been investigated in [2] - [8]. In this paper we give additional properties of the adjacent eccentric distance sum index for simple connected graphs.…”