Secondary K-Range Symmetric Neutrosophic Fuzzy Matrices

Abstract

This paper introduces and explores the concept of secondary k-Range Symmetric (RS) Neutrosophic Fuzzy Matrices (NFM) and establishes its properties and relationships with other symmetric and secondary symmetric NFMs. The study defines secondary k-RS NFMs and provides insightful numerical examples to illustrate their characteristics. The paper investigates the interconnections among s-k-RS, s-RS, k-RS, and RS NFMs, discuss on their mutual relations. Additionally, the necessary and sufficient conditions for a given NFM to qualify as a s-k-RS NFM are identified. The research demonstrates that k-symmetry implies k-RS, and vice versa, contributing to a comprehensive understanding between different types of symmetries in NFMs. Graphical representations of RS, column symmetric, and kernel symmetric adjacency and incidence NFMs are presented, unveiling intriguing patterns and relationships. While every adjacency NFM is symmetric, range symmetric, column symmetric, and kernel symmetric, the incidence matrix satisfies only kernel symmetric conditions. The study further establishes that every range symmetric adjacency NFM is a kernel symmetric adjacency NFM, though the converse does not hold in general. The existence of multiple generalized inverses of NFMs in Fn is explored, with additional equivalent conditions for certain g-inverses of s-κ-RS NFMs to retain the s-κ-RS property. We conclude by characterizing the generalized inverses belonging to specific sets {1, 2}, {1, 2, 3}, and {1, 2, 4} of s-k-RS NFMs, providing a comprehensive framework for understanding the structure and properties of secondary k-Range Symmetric Neutrosophic Fuzzy Matrices. This research contributes to the mathematical literature by introducing a novel class of NFMs and establishing their fundamental properties and relationships, presenting new perspectives on matrix theory in the context of neutrosophic fuzzy logic.