A lower bound on the eccentric connectivity index of a graph

In pharmaceutical drug design, an important tool is the prediction of physicochemical, pharmacological and toxicological properties of compounds directly from their structure. In this regard, the Wiener index, first defined in 1947, has been widely researched, both for its chemical applications and mathematical properties. Many other indices have since been considered, and in 1997, Sharma, Goswami and Madan introduced the eccentric connectivity index  , which has been identified to give a high degree of predictability. If GG is a connected graph with vertex set VV, then the eccentric connectivity index of GG, ξC(G)ξC(G), is defined as ∑v∈Vdeg(v)ec(v), where deg(v) is the degree of vertex vv and ec(v) is its eccentricity. Several authors have determined extremal graphs, for various classes of graphs, for this index. We show that a known tight lower bound on the eccentric connectivity index for a tree TT, in terms of order and diameter, is also valid for a general graph GG, of given order and diameter.