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

Abstract

Background: A topological index of a molecular graph is the numeric quantity that 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 Xu’s results and show that the same graph transformations increase or decrease the first and second general multiplicative Zagreb indices for (Formula presented). As an application, the extremal acyclic, unicyclic, and bicyclic graphs are presented for general multiplicative Zagreb indices. Conclusion: By applying the transformation, we investigated which graphs in the class of acyclic, unicyclic and bicyclic graphs, give the minimum and the maximum general multiplicative Zagreb indices.