On the minimal eccentric connectivity indices of graphs

Let GG be a simple connected graph. The eccentric connectivity index ξc(G)ξc(G) of GG is defined as ξc(G)=∑v∈V(G)d(v)ecG(v)ξc(G)=∑v∈V(G)d(v)ecG(v), where the eccentricity ecG(v)ecG(v) is the largest distance between vv and any other vertex uu of GG. In this paper, we obtain lower bounds on ξc(G)ξc(G) in terms of the number of edges among nn-vertex connected graphs with given diameter. Over connected graphs on nn vertices with mm edges and diameter at least ss, and connected graphs on nn vertices with diameter at least dd, we provide lower bounds on ξc(G)ξc(G) and characterize the extremal graphs, respectively.