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<p>Fractional graph theory generalizes classical spectral methods by introducing nonlocal interactions through fractional powers of the graph Laplacian, <i>L</i><sup>α</sup> for <i>α </i>∈ (0, 1]. This framework interpolates between local and global connectivity, producing dynamics that enhance mixing, robustness, and spectral indistinguishability. In this work, we develop a rigorous analysis of fractional expanders, including explicit constructions, eigenvalue computations, and diffusion behavior We demonstrate that Ramanujan graphs retain their optimality under fractional operators and exhibit improved spectral conditioning, making them, particularly, suitable for fast information dissemination and secure quantum communication. Numerical simulations on regular and Erdős–Rényi graphs illustrate the effects of varying <i>α</i> on spectral gaps and diffusion norms. We further explore applications to consensus dynamics, pseudorandomness, and quantum cryptography, showing that fractional expanders can reduce adversarial inference and enhance key distribution security. The results establish fractional graph operators as a mathematically principled tool for both theoretical and applied network design.</p>

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Fractional graph expanders and network dynamics: spectral properties and diffusion with applications to quantum cryptography

  • 期刊论文

  • 发表日期：

    2026年5月18日

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  • Rami Ahmad El-Nabulsi,
  • Waranont Anukool

摘要

Fractional graph theory generalizes classical spectral methods by introducing nonlocal interactions through fractional powers of the graph Laplacian, Lα for α ∈ (0, 1]. This framework interpolates between local and global connectivity, producing dynamics that enhance mixing, robustness, and spectral indistinguishability. In this work, we develop a rigorous analysis of fractional expanders, including explicit constructions, eigenvalue computations, and diffusion behavior We demonstrate that Ramanujan graphs retain their optimality under fractional operators and exhibit improved spectral conditioning, making them, particularly, suitable for fast information dissemination and secure quantum communication. Numerical simulations on regular and Erdős–Rényi graphs illustrate the effects of varying α on spectral gaps and diffusion norms. We further explore applications to consensus dynamics, pseudorandomness, and quantum cryptography, showing that fractional expanders can reduce adversarial inference and enhance key distribution security. The results establish fractional graph operators as a mathematically principled tool for both theoretical and applied network design.