On eccentric distance sum and minimum degree

Let GG be a connected graph of order nn and minimum degree delta greater than or equal to 2 delta greater than or equal to 2. The eccentric distance sum xi super(d)(G) xi d(G) of GG is defined as capital sigma v[isin]V(G)ecG(v)DG(v), where ecG(v) is the eccentricity of vertex vv in GG and D sub(G)(v)DG(v) is the sum of all distances from vv to other vertices of GG. We prove the upper bound xi d(G) less than or equal to 3 times 5225( delta +1)2n4+O(n3). Our bound is, for a fixed delta delta , asymptotically sharp and it extends a result of Ilic, Yu and Feng (2011), and that of Zhang and Li (2011).