Some further results on the eccentric distance sum

Abstract

Let G = ( V ( G ) , E ( G ) ) be a simple connected graph. Then the eccentric distance sum of G , which is a novel graph invariant by offering a great potential for structure activity/property relationships, is defined as ξ d ( G ) = ∑ v ∈ V ε G ( v ) D G ( v ) , where ε G ( v ) is the eccentricity of the vertex v , and D G ( v ) is the sum of all distance from the vertex v . As a continuation to the parts of [22] (Li et al., J. Math. Anal. Appl. 43:1149–1162, 2015), and [17] (Hua et al., Discrete Appl. Math. 160:170–180, 2012), this paper answers one of some remaining problems in [22] is how to determine the bipartite graphs of even diameter with the minimum EDS, and gives a Nordhaus–Gaddum bound for eccentric distance sum, which is the generalization of the corresponding results in [17] . Moreover, some sharp lower bounds on the EDS of Halin graphs, and of triangle-free and quadrangle-free graphs are respectively presented.