Further results on Zagreb eccentricity coindices

Abstract

The eccentric connectivity index and second Zagreb eccentricity index are well-known graph invariants defined as the sums of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. The coindices of these invariants have recently been proposed by considering analogous contributions from the pairs of non-adjacent vertices. Here, we obtain several lower and upper bounds on the eccentric connectivity coindex and second Zagreb eccentricity coindex in terms of some graph parameters such as order, size, number of non-adjacent vertex pairs, radius, and diameter, and relate these invariants to some well-known graph invariants such as Zagreb indices and coindices, status connectivity indices and coindices, ordinary and multiplicative Zagreb eccentricity indices, Wiener index, degree distance, total eccentricity, eccentric connectivity index, second eccentric connectivity index, and eccentric-distance sum. Moreover, we compute the values of these coindices for two graph constructions, namely, double graphs and extended double graphs.