Extremal regular graphs for the eccentric connectivity index

Abstract

Topological indices are useful tools for identifying properties of chemicals directly from their molecular structure, and are used extensively in pharmaceutical drug design. One such graph-invariant index is the eccentric connectivity index. If G is a connected graph with vertex set V, then the eccentric connectivity index of G, ξ^C (G), is defined as Σ_vεV deg(v) ec(v) where deg(v) is the degree of vertex v and ec(v) is its eccentricity. Current research investigates mathematical properties of this index, and in particular, considers bounds in terms of other parameters. Recently, both Došlić, Saheli and Vukičević [7] and Ilić [12] have stated that it would be interesting to determine extremal graphs with respect to the eccentric connectivity index, for regular (and more specifically, cubic) graphs. When considering such regular graphs, results could equivalently be reformulated in terms of the average eccentricity of the graph. In this note we address this open problem.