Refined-based fundamental pythagorean fuzzy graphs

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, Payame Noor University, Tehran, Iran

Abstract

This paper introduces the notion of refined Pythagorean fuzzy superhypergraphs ($\textnormal{RPFS}$) as a mathematical model for clustering complex data under uncertainty. We define a fundamental relation $\beta$ on an RPFS and prove that its quotient is a Pythagorean fuzzy graph ($\textnormal{PFG}$). We also prove that every $\textnormal{PFG}$ can be embedded in an $\textnormal{RPFS}$, and any $\textnormal{RPFS}$ is a quotientable $\textnormal{PFG}$. A combinatorial formula is proven for the number of distinct $\textnormal{RPFS}$s on any given set and for complete-based $\textnormal{PFG}$s. This study presents the link-degree and superedge-degree points and investigates the relation of link-degree in fundamental $\textnormal{PFG}$ and superedge-degree in any related $\textnormal{RPFS}$. Furthermore, it is analyzed that the direct product of two $\textnormal{RPFS}$s is again an $\textnormal{RPFS}$, and its quotient equals the direct product of their quotients. In the categorical property, it is established that the set of all $\textnormal{RPFS}$s forms a commutative monoid under the direct product operation. Finally, we considered the relation between fuzzy data clustering and RPFS, and clustered, modeled, and combined some real-world problems via RPFS and the concept of its direct product.

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References

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Journal of Mahani Mathematical Research

Articles in Press, Accepted Manuscript
Available Online from 09 July 2026

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History

  • Receive Date: 27 January 2026
  • Revise Date: 17 June 2026
  • Accept Date: 07 July 2026

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