Minimum Eccentric Connectivity Index for Graphs with Fixed Order and Fixed Number of Pending Vertices
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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 pending vertices.
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