Extremal (n,m)-Graphs w.r.t General Multiplicative Zagreb Indices

Abstract

Background:: A topological index of a molecular graph is the numeric quantity which can predict certain physical and chemical properties of the corresponding molecule. Xu et al. introduced some graph transformations which increase or decrease the first and second multiplicative Zagreb indices and proposed a unified approach to characterize extremal (n, m)- graphs. Method:: Graph transformations are used to find the extremal graphs, these transformations either increase or decrease the general multiplicative Zagreb indices. By applying the transformations which increase the general multiplicative Zagreb indices we find the graphs with maximal general multiplicative Zagreb indices and for minimal general Zagreb indices we use the transformations which decrease the index. Result:: In this paper, we extend the Xu’s results and show that the same graph transformations increase or decrease the first and second general multiplicative Zagreb indices for . As an application, the extremal acyclic, unicyclic and bicyclic graphs are presented for general multiplicative Zagreb indices. Conclusion:: By applying the transformation we investigated that in the class of acyclic, unicyclic and bicyclic graphs, which graph gives the minimum and the maximum general multiplicative Zagreb indices.