A short and unified proof of Yu et al.ʼs two results on the eccentric distance sum

Abstract

The eccentric distance sum (EDS) is a novel topological index that offers a vast potential for structure activity/property relationships. For a connected graph G, the eccentric distance sum is defined as ξ d ( G ) = ∑ v ∈ V ( G ) ec G ( v ) D G ( v ) , where ec G ( v ) is the eccentricity of a vertex v in G and D G ( v ) is the sum of distances of all vertices in G from v. More recently, Yu et al. [G. Yu, L. Feng, A. Ilić, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl. 375 (2011) 99–107] proved that for an n-vertex tree T, ξ d ( T ) ⩾ 4 n 2 − 9 n + 5 , with equality holding if and only if T is the n-vertex star S n , and for an n-vertex unicyclic graph G, ξ d ( G ) ⩾ 4 n 2 − 9 n + 1 , with equality holding if and only if G is the graph obtained by adding an edge between two pendent vertices of n-vertex star. In this note, we give a short and unified proof of the above two results.