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<p>Classical (crisp) graph models fail to capture uncertainty inherent in real-world networks, whereas fuzzy topological indices provide a richer description but are computationally expensive. This study investigates whether fuzzy topological indices can be accurately predicted from their crisp counterparts using machine learning. We derive exact closed-form expressions for both crisp and fuzzy first and second Zagreb, Randic, and harmonic indices for two important regular lattices: hexagonal networks <InlineEquation ID="IEq1"><EquationSource Format="TEX">\(HX(n)\)</EquationSource></InlineEquation> and honeycomb networks <InlineEquation ID="IEq2"><EquationSource Format="TEX">\(HC(n)\)</EquationSource></InlineEquation>. Using these expressions, we generate paired datasets and apply linear regression to model the relationship between crisp and fuzzy indices. The results show near-perfect predictive accuracy (<InlineEquation ID="IEq3"><EquationSource Format="TEX">\(R^2 &gt; 0.999\)</EquationSource></InlineEquation> for all models) with very low standard errors and high statistical significance. Our main contribution is a computationally efficient method to estimate fuzzy topological indices without explicit fuzzy graph computations, achieved through simple linear equations that use only easily obtained crisp indices. This work bridges deterministic graph theory, fuzzy uncertainty modelling, and machine learning, offering practical benefits for epidemiology, smart city networks, and nanotechnology.</p>

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Predicting fuzzy topological indices from crisp indices in hexagonal and honeycomb networks using linear regression

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月8日

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  • Jie Qin,
  • Shamaila Yousaf,
  • Mamoona Waris,
  • Anisa Naeem,
  • Keneni Abera Tola

摘要

Classical (crisp) graph models fail to capture uncertainty inherent in real-world networks, whereas fuzzy topological indices provide a richer description but are computationally expensive. This study investigates whether fuzzy topological indices can be accurately predicted from their crisp counterparts using machine learning. We derive exact closed-form expressions for both crisp and fuzzy first and second Zagreb, Randic, and harmonic indices for two important regular lattices: hexagonal networks \(HX(n)\) and honeycomb networks \(HC(n)\). Using these expressions, we generate paired datasets and apply linear regression to model the relationship between crisp and fuzzy indices. The results show near-perfect predictive accuracy (\(R^2 > 0.999\) for all models) with very low standard errors and high statistical significance. Our main contribution is a computationally efficient method to estimate fuzzy topological indices without explicit fuzzy graph computations, achieved through simple linear equations that use only easily obtained crisp indices. This work bridges deterministic graph theory, fuzzy uncertainty modelling, and machine learning, offering practical benefits for epidemiology, smart city networks, and nanotechnology.