Fractal Graphs and their Properties

Perepelitsa, V. A.; Pinchuk, V. P.; Sergeeva, L. N.; Pozdnjakova, A. J.

The idea of representing urban structure and various communication systems (water and energy supply, telephone and cable TV networks) as fractal objects is not absolutely new. However, known works, devoted to this problem use models and approaches from fractal physics. For example, to simulate urban growth Diffusion Limited Aggregation (DLA) model and Dielectric Breakdown (DB) model are used. This study introduces a different approach. Net structure of communication system is described by a graph of special type called regular G(l,r,n)-graph. Authors provide description of such graph, develop iterative process for its generation and prove its self-similarity, i.e. that every regular graph is a pre-fractal. After the infinite number of steps this process generates a fractal. The devised algorithm for generation and grathical representation of regular G(l,r,n)-graphs with different values of l,r and n has been programmed to receive computer simulations. For optimal graphic presentation of pre-fractals the Optimal Space Ordering method was suggested. It is based on the minimization of the >graph energy< value about vertices' coordinates. The effective procedure for optimization was developed that takes into account specific properties of graph energy as objective function For the fractal graph introduced the Hausdorff-Besikovich and similarity dimensions were calculated. It has been shown that >graph energy< is directly related to the graph's fractal properties. For G(3,3,n) and G(4,4,n) graphs fractal dimensions calculated by different methods are the same (D=1,5 and D=2 respectively), while topological dimension of both graphs is 1.

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Perepelitsa, V. A. / Pinchuk, V. P. / Sergeeva, L. N. / et al: Fractal Graphs and their Properties. 2005.

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