# Article

 Title: Symmetry of iteration graphs (English) Author: Carlip, Walter Author: Mincheva, Martina Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 58 Issue: 1 Year: 2008 Pages: 131-145 Summary lang: English . Category: math . Summary: We examine iteration graphs of the squaring function on the rings $\mathbb{Z}/n\mathbb{Z}$ when $n = 2^{k}p$, for $p$ a Fermat prime. We describe several invariants associated to these graphs and use them to prove that the graphs are not symmetric when $k=3$ and when $k\ge 5$ and are symmetric when $k = 4$. (English) Keyword: digraph Keyword: iteration digraph Keyword: quadratic map Keyword: tree Keyword: cycle MSC: 05C20 MSC: 05C62 MSC: 11T99 idZBL: Zbl 1174.05048 idMR: MR2402530 . Date available: 2009-09-24T11:54:02Z Last updated: 2016-04-07 Stable URL: http://hdl.handle.net/10338.dmlcz/128250 . Reference: [1] Earle L. Blanton, Jr., Spencer P. Hurd and Judson S. McCranie: On a digraph defined by squaring modulo $n$.Fibonacci Quart. 30 (1992), 322–334. MR 1188735 Reference: [2] Guy Chassé: Combinatorial cycles of a polynomial map over a commutative field.Discrete Math. 61 (1986), 21–26. MR 0850926, 10.1016/0012-365X(86)90024-5 Reference: [3] John Ellson, Emden Gansner, Lefteris Koutsofios, Stephen C. North and Gordon Woodhull: Graphviz-open source graph drawing tools.Graph drawing (Petra Mutzel, Michael Jünger, and Sebastian Leipert, eds.), Lecture Notes in Computer Science, vol. 2265, Springer-Verlag, Berlin, 2002, Selected papers from the 9th International Symposium (GD 2001) held in Vienna, September 23–26, 2001, pp. 483–484. (English) MR 1962414 Reference: [4] : The GAP Group, Gap-groups, algorithms, and programming, version 4.4, 2005, (http://www.gap-system.org).. Reference: [5] Thomas D. Rogers: The graph of the square mapping on the prime fields.Discrete Math. 148 (1996), 317–324. MR 1368298, 10.1016/0012-365X(94)00250-M Reference: [6] Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory.Czechoslovak Math. J. 54 (2004), 465–485. MR 2059267, 10.1023/B:CMAJ.0000042385.93571.58 Reference: [7] L. Szalay: A discrete iteration in number theory.BDTF Tud. Közl. 8 (1992), 71–91. Zbl 0801.11011 Reference: [8] Troy Vasiga and Jeffrey Shallit: On the iteration of certain quadratic maps over $\text{GF}(p)$.Discrete Math. 277 (2004), 219–240. MR 2033734 .

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