Further results on the eccentric distance sum

Abstract

The eccentric distance sum (EDS) is a novel graph invariant which can be used to predict biological and physical properties, and has a vast potential in structure activity/property relationships. For a connected graph G , its EDS is defined as ξ d ( G ) = ∑ v ∈ V ( G ) e c c G ( v ) D G ( v ) , where e c c 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 . In this paper, we obtain some further results on EDS. We first give some new lower and upper bounds for EDS in terms of other graph invariants. Then we present two Nordhaus–Gaddum-type results for EDS. Moreover, for a given nontrivial connected graph, we give explicit formulae for EDS of its double graph and extended double cover, respectively. Finally, for all possible k values, we characterize the graphs with the minimum EDS within all connected graphs on n vertices with k cut edges and all graphs on n vertices with edge-connectivity k , respectively.