Geodesic
Symmetry of iteration graphs
Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 131-145 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $k = 4$.
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 $k = 4$.
Classification : 05C20, 05C62, 11T99
Keywords: digraph; iteration digraph; quadratic map; tree; cycle 
@article{CMJ_2008_58_1_a8,
     author = {Carlip, Walter and Mincheva, Martina},
     title = {Symmetry of iteration graphs},
     journal = {Czechoslovak Mathematical Journal},
     pages = {131--145},
     year = {2008},
     volume = {58},
     number = {1},
     mrnumber = {2402530},
     zbl = {1174.05048},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a8/}
}
TY  - JOUR
AU  - Carlip, Walter
AU  - Mincheva, Martina
TI  - Symmetry of iteration graphs
JO  - Czechoslovak Mathematical Journal
PY  - 2008
SP  - 131
EP  - 145
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a8/
LA  - en
ID  - CMJ_2008_58_1_a8
ER  - 
%0 Journal Article
%A Carlip, Walter
%A Mincheva, Martina
%T Symmetry of iteration graphs
%J Czechoslovak Mathematical Journal
%D 2008
%P 131-145
%V 58
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a8/
%G en
%F CMJ_2008_58_1_a8
Carlip, Walter; Mincheva, Martina. Symmetry of iteration graphs. Czechoslovak Mathematical Journal, Tome 58 (2008) no. 1, pp. 131-145. http://geodesic.mathdoc.fr/item/CMJ_2008_58_1_a8/

[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

[2] Guy Chassé: Combinatorial cycles of a polynomial map over a commutative field. Discrete Math. 61 (1986), 21–26. | DOI | MR

[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

[4] The GAP Group, Gap-groups, algorithms, and programming, version 4.4, 2005, ( http://www.gap-system.org)</b>

[5] Thomas D. Rogers: The graph of the square mapping on the prime fields. Discrete Math. 148 (1996), 317–324. | DOI | MR

[6] Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory. Czechoslovak Math. J. 54 (2004), 465–485. | DOI | MR

[7] L. Szalay: A discrete iteration in number theory. BDTF Tud. Közl. 8 (1992), 71–91. | Zbl

[8] Troy Vasiga and Jeffrey Shallit: On the iteration of certain quadratic maps over $\text{GF}(p)$. Discrete Math. 277 (2004), 219–240. | MR