Ordering the Zagreb coindices of connected graphs

In this paper, we give a simple approach to order the first Zagreb indices of connected graphs and the second Zagreb coindices of trees and unicyclic graphs, respectively. As an application of our new method, we determine the first eight smallest and the first three largest (respectively, first eight smallest and first three largest, first seven smallest and first two largest) values of the first Zagreb coindices in the class of trees (respectively, unicyclic graphs, bicyclic graphs) on n vertices, and we also determine the first eleven (respectively, thirteen) smallest values of the second Zagreb coindices in the class of trees (respectively, unicyclic graphs) on n vertices. Furthermore, we also identify the smallest value of the first Zagreb coindices in the class of chemical trees on n ≥ 8 vertices, partially giving an answer to a question of Ashrafi, Došlić and Hamzeh.