A comprehensive study on fuzzy and crisp graph indices: generalized formulae, proximity and accuracy analysis

Muhammad Umar Mirza; Rukhshanda Anjum; Maged Z. Youssef; Turki Alsuraiheed; Muhammad Umar Mirza; Rukhshanda Anjum; Maged Z. Youssef; Turki Alsuraiheed

doi:10.3934/math.20231582

AIMS Mathematics

2023, Volume 8, Issue 12: 30922-30939. doi: 10.3934/math.20231582

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Research article Special Issues

A comprehensive study on fuzzy and crisp graph indices: generalized formulae, proximity and accuracy analysis

1.
Department of Mathematics and Statistics, University of Lahore, Lahore, Pakistan
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University(IMSIU), Riyadh, Saudi Arabia
3.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 16 August 2023 Revised: 04 November 2023 Accepted: 09 November 2023 Published: 17 November 2023
MSC : 05C72

This article presents a set of generalized $ \wp $-dependent formulae for fuzzy and crisp versions of various graph indices, including the first and second Zagreb indices, harmonic index and Randic index. These formulae are applied to the identity graph of the commutative ring $ Z_{\wp} $, and the resulting indices are calculated by using MATLAB software for 20 prime numbers. The generated data were applied for machine learning by using Python and Jupyter notebook to investigate the relationship between fuzzy and crisp indices. The article also includes the relationship between fuzzy and crisp indices in the form of six-degree polynomials and an error analysis.
- fuzzy graph,
- topological indices,
- fuzzy indices,
- polynomial regression,
- machine learning,
- identity graph
Citation: Muhammad Umar Mirza, Rukhshanda Anjum, Maged Z. Youssef, Turki Alsuraiheed. A comprehensive study on fuzzy and crisp graph indices: generalized formulae, proximity and accuracy analysis[J]. AIMS Mathematics, 2023, 8(12): 30922-30939. doi: 10.3934/math.20231582

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Abstract

This article presents a set of generalized $ \wp $-dependent formulae for fuzzy and crisp versions of various graph indices, including the first and second Zagreb indices, harmonic index and Randic index. These formulae are applied to the identity graph of the commutative ring $ Z_{\wp} $, and the resulting indices are calculated by using MATLAB software for 20 prime numbers. The generated data were applied for machine learning by using Python and Jupyter notebook to investigate the relationship between fuzzy and crisp indices. The article also includes the relationship between fuzzy and crisp indices in the form of six-degree polynomials and an error analysis.

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