Revisiting Tripolar HyperNeutrosophic Graphs and Their Applications

Abstract

Graph-based models provide a fundamental mathematical framework for representing relational systems through vertices and their connections. Fuzzy set theory extends classical set theory by assigning each element a membership degree in [0ⓜ,1], thereby allowing partial rather than binary membership. Among the various extensions developed for uncertainty modeling, Neutrosophic Sets and HyperNeutrosophic Sets provide flexible frameworks for representing truth, indeterminacy, and falsity, including set-valued and higher-order forms of uncertainty. Motivated by these developments, this paper investigates Tripolar HyperNeutrosophic Sets and Tripolar HyperNeutrosophic Graphs. These frameworks combine the higher-order structure of HyperNeutrosophic Sets and HyperNeutrosophic Graphs with the three-pole representation of Tripolar Neutrosophic Sets, providing a unified setting for relational systems involving multiple perspectives and neutrosophic uncertainty. Since Fuzzy Sets, Neutrosophic Sets, and related models have also been applied in astronomy and astrophysics, we further present brief illustrative applications of Tripolar HyperNeutrosophic Graphs in astronomy.