Connectivity, diameter, minimal degree, independence number and the eccentric distance sum of graphs

The eccentric distance sum (EDS) of a connected graph G is defined as ξd(G)=∑{u,v}⊆VG(εG(u)+εG(v))dG(u,v), where εG(⋅) is the eccentricity of the corresponding vertex and dG(u,v) is the distance between u and v in G. In this paper, some extremal problems on the EDS of an n-vertex graph with respect to other two graph parameters are studied. Firstly, k-connected (bipartite) graphs with given diameter having the minimum EDS are characterized. Secondly, sharp lower bound on the EDS of connected graphs with given connectivity and minimum degree is determined. Lastly sharp lower bound on the EDS of connected graphs with given connectivity and independence number is determined as well.