On eccentric distance sum and minimum degree

Let GG be a connected graph of order nn and minimum degree δ≥2δ≥2. The eccentric distance sum ξd(G)ξd(G) of GG is defined as ∑v∈V(G)ecG(v)DG(v), where ecG(v) is the eccentricity of vertex vv in GG and DG(v)DG(v) is the sum of all distances from vv to other vertices of GG. We prove the upper bound ξd(G)≤3⋅5225(δ+1)2n4+O(n3). Our bound is, for a fixed δδ, asymptotically sharp and it extends a result of Ilić, Yu and Feng (2011), and that of Zhang and Li (2011).