A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for $\alpha \in {\mathbb{R}}^{+}\backslash \left\{1\right\}.$ Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for $\alpha \in {\mathbb{R}}^{+}\backslash \left\{1\right\},$ we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices.