On the minimal eccentric connectivity indices of graphs

Let G be a simple connected graph. The eccentric connectivity index ξ c ( G ) of G is defined as ξ c ( G ) = ∑ v ∈ V ( G ) d ( v ) e c G ( v ) , where the eccentricity e c G ( v ) is the largest distance between v and any other vertex u of G . In this paper, we obtain lower bounds on ξ c ( G ) in terms of the number of edges among n -vertex connected graphs with given diameter. Over connected graphs on n vertices with m edges and diameter at least s , and connected graphs on n vertices with diameter at least d , we provide lower bounds on ξ c ( G ) and characterize the extremal graphs, respectively.