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

Lulu Peng (), Dirong Chen (), Cheng Zeng, Yumei Xue () and Huixia He ()
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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

International Journal of Modern Physics C (IJMPC), 2025, vol. 36, issue 02, 1-21

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.

Keywords: Hierarchical product; average Fermat distance; small-world; Wiener index; Laplacian spectrum (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0129183124501894

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