On the extreme eccentric distance sum of graphs with some given parameters

The eccentric distance sum (EDS) of a connected graph GG is defined as ξd(G)=∑v∈VG(εG(u)+εG(v))dG(u,v)ξd(G)=∑v∈VG(εG(u)+εG(v))dG(u,v), where εG(⋅)εG(⋅) is the eccentricity of the corresponding vertex and dG(u,v)dG(u,v) is the distance between the vertices uu and vv. The eccentric distance sum is a novel graph invariant with vast potential in structure activity/property relationships. In this paper, the relationship between the EDS and some other graph parameters such as the order, the size, the diameter and the connectivity are studied. First, the sharp lower bound on the EDS of nn-vertex connected triangle-free graphs is determined. Second, a sharp lower bound on the EDS of nn-vertex connected graphs of size mm is characterized. Then, a sharp lower bound on the EDS of nn-vertex connected graph with given diameter is determined. At last, a sharp upper bound on the EDS of nn-vertex graph with given (even) connectivity is determined.