Further results on the eccentric distance sum

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 GG, its EDS is defined as ξd(G)=∑v∈V(G)eccG(v)DG(v)ξd(G)=∑v∈V(G)eccG(v)DG(v), where eccG(v)eccG(v) is the eccentricity of a vertex vv in GG and DG(v)DG(v) is the sum of distances of all vertices in GG from vv. 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 kk values, we characterize the graphs with the minimum EDS within all connected graphs on nn vertices with kk cut edges and all graphs on nn vertices with edge-connectivity kk, respectively.