Comparing Eccentricity-Based Graph Invariants

The first and second Zagreb eccentricity indices (EM1 and EM2), the eccentric distance sum (EDS), and the connective eccentricity index (CEI) are all recently conceived eccentricity-based graph invariants, some of which found applications in chemistry. We prove that EDS ≥ EM1 for any connected graph, whereas EDS > EM2 for trees. Moreover, in the case of trees, EM1 ≥ CEI, whereas EM2> CEI for trees with at least three vertices. In addition, we compare EDS with EM2, and compare EM1, EM2 with CEI for general connected graphs under some restricted conditions.