Topological indices (TIs) are numerical structures that are associated with a graph to identify its topology. TIs are highly popular in the literature with a wide range of applications from chemistry to economics. However, TIs have limitations in representating complex relations within the graphs creating some uncertainities. Fuzzy graph (FG) and intuitionistic fuzzy graph (IFG) are introduced to overcome these uncertainities. While a FG a describes degree of membership of an object in a graph, IFG delineate information on membership or nonmembership under uncertainity. This study aims to introduce novel TIs such as the general second Zagreb index, the Sombor index of the third version, and the Sombor index of the fourth version in the IFG framework in order to improve practicality of FG and IFG applications. Some properties of the proposed indices and their upper bounds are provided as well. Proposed TIs are applied to an internet routing network as a case study. Results of the study show that adding more internet routers in the network can increase internet speed and the strength of the entire system. Finally, comparative studies for the Sombor index of the third version and the Sombor index of the fourth version are also revealed.

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.

Euclidean geometry is utilized to establish the Sombor graph parameter and its invariants. It is sum of all adjacent vertices in graph theory $\sqrt{d_{\mathrm{\Upsilon }}^2+d_{\mathrm{\Gamma }}^2}$ where $d_{\mathrm{\Upsilon }}$ is the degree of the vertex ϒ. Geometrical interpretation is used to describe the new Sombor indices types. We examined, recently developed Sombor indices for various nanotube Y-junctions in this article. In specifically, the first area-based Sombor index was introduced by Euclidean geometry. Angular orientation concept to construct the second, fourth, and sixth Sombor graph parameters, while third and fifth Sombor graph parameters are constructed by perimeter.