A Note on q-Fractional Neutrosophic Sets and Graphs

Takaaki Fujita1,* Ajoy Kanti Das2 Suman Das3 Sankar Prasad Mondal4

1 Independent Researcher, Tokyo, Japan

2 Department of Mathematics, Tripura University, Agartala-799022, Tripura, India

3 Department of Education, National Institute of Technology Calicut, Kozhikode-673601, Kerala, India

4 Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, Haringhata-741249, West Bengal, India

Emails: Takaaki.fujita060@gmail.com · ajoykantidas@gmail.com · dr.sumandas1995@gmail.com · sankar.mondal02@gmail.com

Received: October 07, 2025 Revised: December 10, 2025 Accepted: January 08, 2026 ⋆ Corresponding author

ABSTRACT

A neutrosophic set assigns to each element three (generally independent) degrees: truth, indeterminacy, and falsity.

A q-fractional fuzzy set assigns membership and nonmembership degrees in [0,1] to each element, constrained by

(μ +ν)/q ≤ 1 for q ≥ 2. A q-fractional fuzzy graph is a graph whose vertices and edges carry q-fractional fuzzy

degrees, with edge membership bounded by endpoints and edge nonmembership dominating them. In this paper, we

introduce q-fractional neutrosophic sets and graphs as an extension of q-fractional fuzzy sets and q-fractional fuzzy

graphs, and we investigate their fundamental properties.

Keywords: Neutrosophic Graph q-fractional fuzzy set q-fractional fuzzy graph

1. INTRODUCTION

Classical (crisp) set theory encodes membership in a binary

way and, by design, does not directly accommodate

vagueness, partial truth, or inconsistent evidence. To

model graded information, Zadeh introduced fuzzy sets in

1965 [1]. Atanassov subsequently proposed intuitionistic

fuzzy sets [2, 3], in which membership and non-membership

degrees are specified separately and the residual part is interpreted

as hesitation. Smarandache further extended this

line of research to neutrosophic sets [4], assigning to each

element three typically independent degrees of truth, indeterminacy,

and falsity, thereby providing a flexible framework

for representing incomplete, indeterminate, and contradictory

information. Several extensions and related variants of

neutrosophic sets are also known, including Bipolar Neutrosophic

Sets[5, 6], HyperNeutrosophic Sets[7], and m-polar

Neutrosophic Sets. Because of their theoretical and practical

significance, neutrosophic sets and their extensions have been

extensively studied in numerous recent publications.

Uncertainty-aware semantics have also been incorporated

into network models. Fuzzy graphs attach values in [0,1] to

vertices and edges [8, 9], while intuitionistic fuzzy graphs

enrich this framework by adding explicit non-membership

information [10, 11]. Neutrosophic graphs further include an

indeterminacy component, enabling a finer representation of

ambiguity in relational data [12, 13, 14]. Neutrosophic graphs

have been actively studied in various recent works [15, 16].

More recently, plithogenic graphs have been proposed to

quantify contradiction effects across multiple attributes [17].

For clarity, Table 1 summarizes the main differences between

fuzzy graphs and neutrosophic graphs.

Along a different axis of generalization, q-fractional models

relax the usual normalization constraints by introducing a

parameter q ≥ 2. A q-fractional fuzzy set assigns membership

and non-membership degrees in [0,1] to each element,

subject to the scaled constraint (μ +ν)/q ≤ 1[18, 19, 20].

Analogously, a q-fractional fuzzy graph is a graph whose

vertices and edges carry q-fractional fuzzy degrees, where

edge membership is bounded by its endpoints and edge non-