Concepts such as fuzzy sets, neutrosophic sets, rough sets, and plithogenic sets have been extensively studied as formal tools for modeling uncertainty, and they have found broad applications across many disciplines. A Double-Valued Neutrosophic Set (DVNS) extends the classical neutrosophic framework by splitting indeterminacy into two distinct components: one leaning toward truth and the other leaning toward falsity. In recent years, further refinements—namely Triple-Valued, Quadruple-Valued, and Quintuple-Valued Neutrosophic Sets—have also been introduced and investigated. These uncertainty models have naturally been lifted to graph-theoretic settings, where vertices and edges represent entities and relationships under ambiguity. Although fuzzy graphs and neutrosophic graphs have been widely studied, there are practical situations in which the ambiguity emphasized in neutrosophic graphs must be classified and analyzed more finely. Motivated by this need, and in order to bridge the above gap, this paper defines Double-Valued, Triple-Valued, Quadruple-Valued, and Quintuple-Valued Neutrosophic Graphs and investigates their fundamental properties.
Fujita et al. (2026) studied this question.