Uncertain Labeling Graphs and Uncertain Graph Classes (with Survey for Various Uncertain Sets)

Abstract

Graph theory, a branch of mathematics, studies the relationships between entities using vertices and edges. Uncertain Graph Theory has emerged within this field to model the uncertainties present in real-world networks. Graph labeling involves assigning labels, typically integers, to the vertices or edges of a graph according to specific rules or constraints. This paper introduces the concept of the Turiyam Neutrosophic Labeling Graph, which extends the traditional graph framework by incorporating four membership values—truth, indeterminacy, falsity, and a liberal state—at each vertex and edge. This approach enables a more nuanced representation of complex relationships. Additionally, we discuss the Single-Valued Pentapartitioned Neutrosophic Labeling Graph.The paper also examines the relationships between these novel graph concepts and other established types of graphs. In the Future Directions section, we propose several new classes of Uncertain Graphs and Labeling Graphs. And the appendix of this paper details the findings from an investigation into set concepts within Uncertain Theory. These set concepts have inspired numerous proposals and studies by various researchers, driven by their applications, mathematical properties, and research interests.