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Minimum eccentric connectivity index for graphs with fixed order and fixed number of pendant vertices

Devillez Gauvain (Computer Science Department - Algorithms Lab, University of Mons, Mons, Belgium)
Hertz Alain (Department of Mathematics and Industrial Engineering, Polytechnique Moontréal - Gerad, Montréal, Canada)
Mélot Hadrien (Computer Science Department - Algorithms Lab, University of Mons, Mons, Belgium)
Hauweele Pierre (Computer Science Department - Algorithms Lab, University of Mons, Mons, Belgium)

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.

Keywords: Extremal Graph Theory, Eccentric Connectivity Index, Pendant Vertices