Graph Decomposition Techniques in Neutrosophic Zero Divisor Models of Commutative Ring

Abstract

This paper introduces the framework of neutrosophic zero divisor graphs as an extension of fuzzy
 zero divisor graphs. Fuzzy zero divisor graphs capture partial uncertainty through membership degrees but
 do not fully account for indeterminacy. To address this limitation, we incorporate the neutrosophic member
ship triplet (T ,I,F), which represents truth, indeterminacy, and falsity, into the algebraic graph-theoretic
 structure of commutative rings.
 We focus on rings of the form Zn, particularly for n = 2p,3p,5p,pq, and the general case kp, where p
 and q are primes. A central contribution is a decomposition theorem showing that the NZDG ΓN (Zkp) can
 be expressed as the disjoint union KN
 1,p−1 ∪ KN
 p−1,k−2. This result generalizes earlier fuzzy-based studies and
 provides a unified structural framework for analyzing uncertainty in algebraic systems.
 Our comparative analysis shows that neutrosophic modeling explicitly handles indeterminacy, giving it clear
 advantages over fuzzy and intuitionistic fuzzy approaches. The proposed methodology is explained step by
 step, with attention to its rationale, scope, and limitations. Potential applications include algebraic modeling,
 logic-based network design, and soft computing. We also outline limitations and future research directions,
 emphasizing the role of NZDGs in advancing the study of uncertainty in algebraic structures.

DOI: 10.5281/zenodo.17114555