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Article title MODELING FRACTAL STRUCTURES GROWTH BY MEANS SIMPLEST PREFRACTAL GRAPHS
Authors A.M. Kochkarov, A.N. Kochkarova, L.H. Khapaeva
Section SECTION I. TECHNOLOGY MANAGEMENT AND MODELING
Month, Year 02, 2016 @en
Index UDC 519.1
DOI
Abstract Development of structures happens in the self-organized systems on the basis of certain principles. One of such fundamental principles is fractality of development. In relation to graphs this principle is realized by the algorithm leading to creation of sequence of prefractal graphs. The essence of this process consists in replacement of node of the graphs (of its primary element) with new structure (graphs) by certain rules. This work investigates properties of the elementary prefractal graphs (further called the elementary prefractal trees EPT), and the algorithm in EPT: "with each stage of the replacement of node with a priming (RNP) all nodes of the previous prefractal tree are updated, and their quantity doubles. At the same time, the set of edges with each stage of RNP "accumulates ", the edge of the first rank is supplemented with two edges of the second, then four edges of the third and so on". Proved results: 1) in the elementary prefractal trees all edges incidental to the same node are edges of different ranks; 2) if the tree has a perfect paracombination, then it unique; 3) set of edges of the L rank of the elementary prefractal tree of forms the edge covering of this tree that is a perfect paracombination; 4) a unique trajectory corresponds to and elementary prefractal tree. Thus the theorem is proved that at the considered transfer there are no isomorphic root trees. Isomorphism at such approach is excluded, because first, splitting a root tree on an edge of balance leads to two components isomorphically not comparable as one of them contains a root of an initial tree, and secondly symmetry which regard nodes that the configuration joins is excluded.

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Keywords Priming; the elementary prefractal trees; paracombination; configuration.
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