Relations between Zagreb coindices and some distance-based topological indices

For a nontrivial graph G, its first Zagreb coindex is defined as the sum of degree sum over all non-adjacent vertex pairs in G and the second Zagreb coindex is defined as the sum of degree product over all non-adjacent vertex pairs in G. Till now, established results concerning Zagreb coindices are mainly related to composite graphs and extremal values of some special graphs. The existing literatures witnessed no results dealing with the relations between Zagreb coindices and distance-based topological indices so far. Aiming at filling in this gap, we reveal the relations between the first Zagreb coindex and some distance-based topological indices here. We establish sharp bounds on the first Zagreb coindex in terms of distance-based topological indices including Wiener index, eccentric connectivity index, eccentric distance sum, degree distance and reverse degree distance.