Scaling Limits and Fractal Dimensions of $ω$-Substitution Graph Systems

Abstract

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.