Minimum Eccentric Connectivity Index for Graphs with Fixed Order and Fixed Number of Pendant Vertices

Abstract

The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product d_G(v)e_G(v), where d_G(v) is the degree of v in G and e_G(v) is the maximum distance between v and any other vertex of G. This index is helpful for the prediction of biological activities of diverse nature, a molecule being modeled as a graph where atoms are represented by vertices and chemical bonds by edges. We characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order n. Also, 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.