On Reformulated Zagreb Indices with Respect to Acyclic, Unicyclic and Bicyclic Graphs

The authors Milicevic et al. introduced the reformulated Zagreb indices [19], which is a generalization of classical Zagreb indices of chemical graph theory. in the paper, we characterize the extremal properties of the first reformulated Zagreb index. We first introduce some graph operations which increase or decrease this index. Furthermore, we will determine the extremal acyclic and bicyclic graphs with minimum and maximum of the first Zagreb index by a unified method, respectively. Recently, Ilia and Zhou [18] characterized the extremal graph of unicyclic graphs with the first reformulated Zagreb index. We will provide a shorter proof.