Geodesic
On attractors in finite dynamic systems of complete graphs orientations
Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 112-114 Cet article a éte moissonné depuis la source Math-Net.Ru

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Finite dynamic systems of complete graphs orientations are considered. The states of such a system $S=(\Gamma_{K_n},\alpha)$, $n>1$, are all possible orientations $G$ of the complete graph $K_n$, and evolutionary function $\alpha$ transforms a given state $G$ by reversing all arcs in $G$ that enter into sinks, and there are no other differences between the given ($G$) and the next ($\alpha(G)$) states. The following criterion for belonging states to attractors in $S$ is given: a state $G$ belongs to an attractor if and only if it hasn't a sink or its indegrees vector is a permutation of numbers $0,1,\dots,n-1$. All attractors in $S$ are the attractors of length $1$, each of which consists of states without sinks, and the attractors of length $n$, each of which consists of states with indegrees vectors being permutations of numbers $0,1,\dots,n-1$. Any such an attractor represents a circuit, for every state $G$ in which if the indegrees vector of $G$ is $(d^-(v_1),d^-(v_2),\dots,d^-(v_n))$, then the indegrees vector of $\alpha(G)$ is $(d^-(v_1)+1,d^-(v_2)+1,\dots,d^-(v_n)+1)$, where the addition is calculated modulo $n$. Note that in system $S$, the number of attractors of length $n$ is equal to $(n-1)!$ and the number of states belonging to them is equal to $n!$.
Keywords: attractor, complete graph, evolutionary function, finite dynamic system, graph
Mots-clés : graph orientation. 
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     title = {On attractors in finite dynamic systems of complete graphs orientations},
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A. V. Zharkova. On attractors in finite dynamic systems of complete graphs orientations. Prikladnaya Diskretnaya Matematika. Supplement, no. 9 (2016), pp. 112-114. http://geodesic.mathdoc.fr/item/PDMA_2016_9_a43/

[1] Bogomolov A. M., Salii V. N., Algebraicheskie osnovy teorii diskretnykh sistem, Nauka, Fizmatlit, M., 1997 | MR

[2] Vlasova A. V., Issledovanie evolyutsionnykh parametrov v dinamicheskikh sistemakh dvoichnykh vektorov, Svidetelstvo o gosudarstvennoi registratsii programmy dlya EVM No 2009614409, vydannoe Rospatentom. Zayavka No 2009613140. Data postupleniya 22 iyunya 2009 g. Zaregistrirovano v Reestre programm dlya EVM 20 avgusta 2009 g.

[3] Vlasova A. V., “Attraktory konechnykh dinamicheskikh sistem, assotsiirovannykh s tsepyami i tsiklami”, Sb. tez. dokl. konf. molodykh uchenykh, Vyp. 1, SPbGU ITMO, SPb., 2011, 70–71

[4] Zharkova A. V., “Kolichestvo attraktorov v dinamicheskikh sistemakh, assotsiirovannykh s tsiklami”, Matem. zametki, 95:4 (2014), 529–537 | DOI | MR | Zbl

[5] Zharkova A. V., “Attraktory v konechnykh dinamicheskikh sistemakh dvoichnykh vektorov, assotsiirovannykh s orientatsiyami palm”, Prikladnaya diskretnaya matematika, 2014, no. 3(25), 58–67

[6] Barbosa V. C., An atlas of edge-reversal dynamics, Chapman Hall/CRC, London, 2001 | MR | Zbl