Scaling Limits and Fractal Dimensions of ω-Substitution Graph Systems

@misc{zeng2026scaling,
  title = {{Scaling Limits and Fractal Dimensions of $ω$-Substitution Graph Systems}},
  author = {Qingcheng Zeng and Yumei Xue and Cheng Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05960},
  eprint = {2611.05960},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05960},
}

TY  - UNPB
TI  - Scaling Limits and Fractal Dimensions of ω-Substitution Graph Systems
AU  - Qingcheng Zeng
AU  - Yumei Xue
AU  - Cheng Zeng
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05960
UR  - https://arxiv.org/abs/2611.05960
AB  - This paper introduces a class of non-homogeneous self-similar constructions, called ω-substitution graph system, in which the interface vertices are determined by an infinite sequence ω. The associated graph sequences naturally form compact metric measure spaces, and under suitable assumptions on ω, we establish their convergence in the Gromov-Hausdorff-Prokhorov sense. We determine the Hausdorff and box-counting dimensions of the limiting space. We further prove that the discrete Hausdorff dimension of the graph sequence coincides with the Hausdorff dimension of its scaling limit. As a direct application of the scaling limit, asymptotic formulas are derived for a class of generalized Wiener indices, which include the classical Wiener index and Kirchhoff index as special cases. Finally, closed-form expressions for these indices are also obtained.
ER  -