A study of a new variant of the eccentric connectivity index for composite graphs

The eccentric connectivity index is a well-known graph invariant defined as the sum of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. Recently, a new variant of the eccentric connectivity index namely multiplicative eccentric connectivity index has been proposed. In this paper, we investigate the behavior of this new invariant under various families of composite graphs. Exact expressions for the values of the multiplicative eccentric connectivity index of sum, disjunction, symmetric difference, lexicographic product, and a special case of strong product, and upper and/or lower bounds for the multiplicative eccentric connectivity index of generalized hierarchical product, Cartesian product, rooted product, corona product, and strong product are presented.