Induced Neutrosophic Topologies from Neutrosophic Graph Coloring: A Rigorous Approach to Approximation Under Indeterminacy

Abstract

This paper initiates and studies a new category of topological structure called neutrosophic topologies, which originate from the neutrosophic coloring of neutrosophic graphs. Expanding upon classical and intuitionistic graph colorings, neutrosophic coloring allots a triple value — truth, indeterminacy, and falsity to each vertex, thereby giving a more refined depiction of uncertainty. Utilizing this enriched coloring model, the authors build a collection of vertex subsets which satisfy the axioms of a topology using neutrosophic set operations. This leads to the development of neutrosophic color topological spaces. Within this framework, two important approximation operators — neutrosophic lower and upper color approximations are defined. These are inspired by rough set theory but enriched through neutrosophic logic. The operators are shown to exhibit crucial topological features such as monotonicity, idempotency, and boundary formation, all based on color-derived neighborhood systems. The paper also discusses the connection between the lower and upper approximations and presents applications in domains like information systems and uncertain network structures, where traditional binary or fuzzy approaches fall short. The work lays a theoretical foundation for unified reasoning in graph theory and topology under indeterminacy, with implications for further research in neutrosophic mathematics, granular computing, and uncertain data analysis.