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

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.