Minimum eccentric connectivity index for graphs with fixed order and fixed number of pendant vertices - Devillez Gauvain; Hertz Alain; Mélot Hadrien; Hauweele Pierre (Yugoslav Journal of Operations Research, 2019)
The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product dG(v)eG(v), where dG(v) is the degree of v in G and eG(v) is the maximum distance between v and any other vertex of G. We characterize, with a new elegant proof, those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order n. Then, given two integers n and p with p ≤ n - 1, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order n with p pendant vertices.