Advances in Neutrosophic Graph Theory: Topological Structures, Neutrosophic Bridges, and Applications via MATLAB

Abstract

As modern systems become increasingly complex, there is a growing need for advanced mathe- matical frameworks capable of modeling uncertainty, vagueness, and indeterminacy. This research presents a novel contribution by integrating neutrosophic set theory with graph theory and topol- ogy, providing a more flexible model for handling complex networked data. The study investigates the fundamental properties of neutrosophic graphs (NG), introducing new types of arcs along with two innovative constructs: the neutrosophic ψ-bridges and Ξ-bridges, offering deeper in- sights into the structural dynamics of neutrosophic graphs. A major advancement in this work is the introduction of the neutrosophic Generalized Adjacency Topological Space (NGATTS), a novel topological framework that redefines classical concepts such as T-closure and T-open sets in the context of graph structures. Through a rigorous series of definitions and theorems, the paper proves that every NGATTS can be represented by a graph and vice versa. To validate the theoretical findings, the neutrosophic graphs and their related structures were implemented and analyzed using MATLAB. The practical experiments involved constructing various well-known graph models such as complete graphs, wheel graphs, cycles, stars, and weak graphs environment, successfully visualizing the new types of arcs and bridges, and confirming their compliance with the NGATTS framework. Overall, this research provides a comprehensive and innovative bridge between topology and graph theory, combining robust theoretical development with practical im- plementation. It opens new directions for the topological analysis of complex networks and offers a valuable reference for researchers in applied mathematics and network science.