A Mathematical Study of Tripolar Intuitionistic Fuzzy Graphs

Abstract
Graph-based models provide a fundamental mathematical framework for representing relational systems in terms of vertices and their connections. Fuzzy sets extend classical set theory by assigning each element a membership degree in the interval [0,1], thereby allowing partial membership rather than restricting membership to binary values. An intuitionistic fuzzy set assigns both membership and non-membership degrees to each element, subject to the condition that their sum does not exceed one, thus explicitly representing the degree of hesitation. Similarly, an intuitionistic fuzzy graph assigns membership and non-membership degrees to vertices and edges, enabling uncertain relationships to be modeled while maintaining compatibility between vertex and edge information. Motivated by these frameworks, this paper investigates Tripolar Intuitionistic Fuzzy Sets and Tripolar Intuitionistic Fuzzy Graphs, which extend intuitionistic fuzzy models by incorporating a tripolar representation of uncertain information.

Keywords: Tripolar Fuzzy Graph, Intuitionistic Fuzzy Set, Tripolar Intuitionistic Fuzzy Set, Tripolar Intuitionistic Fuzzy Graph