| 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:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128250 |
| . |
| Reference:
|
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| 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 |
| . |
