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Fractal version of average Fermat distance on some small-world hierarchical networks

Author

Listed:
  • Lulu Peng

    (School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China)

  • Dirong Chen

    (School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China)

  • Cheng Zeng

    (School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, Shandong Province, 264003, P. R. China)

  • Yumei Xue

    (School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China)

  • Huixia He

    (School of Mathematical Sciences, Beihang University, Beijing 100083, P. R. China)

Abstract

Fermat–Wiener index based on topological Wiener index is the total sum of Fermat distance over all the triplets for vertices. In this paper, we construct a class of hierarchical graphs based on hierarchical product generalized from Cartesian product. We study some critical properties of the hierarchical networks by investigating its topological indices. Applying the finite pattern method, we analytically deduce the dominant term of average Fermat distance and obtain its asymptotic formula, which implies small-world property. We finally exhibit a close connection between Fermat–Wiener index and related graph invariants like average geodesic distance, Wiener index and eigenvalues of Laplacian matrix.

Suggested Citation

  • Lulu Peng & Dirong Chen & Cheng Zeng & Yumei Xue & Huixia He, 2025. "Fractal version of average Fermat distance on some small-world hierarchical networks," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 36(02), pages 1-21, February.
    • Handle: RePEc:wsi:ijmpcx:v:36:y:2025:i:02:n:s0129183124501894
    DOI: 10.1142/S0129183124501894
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