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Fractal dimensions for deterministic and random substitution graph systems. (arXiv:2212.01987v3 [math.DS] UPDATED)

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Based on [19], this paper aims to introduce fractal geometry into graph
theory. To do so, we construct and study the fractal-like graphs by the method
of substitution, called deterministic or random substitution graph systems.
Though the idea of substitution is common in terms of fractal geometry and
dynamical systems, the analysis of fractals regarding graph theory remains an
immature field. By investigating the properties of the systems such as diameter
and distal, we obtain two main results in this paper. The first of which, for
deterministic substitution graph systems, the box-counting dimension and
Hausdorff dimension are analytically coincidental by explicit formulae. Another
finding is, for random substitution graph systems, that we prove almost surely
every graph limit has the same box-counting and Hausdorff dimensions
numerically by their Lyapunov exponents. Through these conclusions, the
substitution graph system will potentially lead to new research directions.

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