Symmetry of iteration graphs

Walter Carlip; Martina Mincheva

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 1, page 131-145
  • ISSN: 0011-4642

Abstract

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We examine iteration graphs of the squaring function on the rings ℤ / n 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 ≥ 5 and are symmetric when k = 4 .

How to cite

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Carlip, Walter, and Mincheva, Martina. "Symmetry of iteration graphs." Czechoslovak Mathematical Journal 58.1 (2008): 131-145. <http://eudml.org/doc/31203>.

@article{Carlip2008,
abstract = {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 $k = 4$.},
author = {Carlip, Walter, Mincheva, Martina},
journal = {Czechoslovak Mathematical Journal},
keywords = {digraph; iteration digraph; quadratic map; tree; cycle; digraph; iteration digraph; quadratic map; tree; cycle},
language = {eng},
number = {1},
pages = {131-145},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Symmetry of iteration graphs},
url = {http://eudml.org/doc/31203},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Carlip, Walter
AU - Mincheva, Martina
TI - Symmetry of iteration graphs
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 1
SP - 131
EP - 145
AB - 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 $k = 4$.
LA - eng
KW - digraph; iteration digraph; quadratic map; tree; cycle; digraph; iteration digraph; quadratic map; tree; cycle
UR - http://eudml.org/doc/31203
ER -

References

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  1. On a digraph defined by squaring modulo n , Fibonacci Quart. 30 (1992), 322–334. (1992) MR1188735
  2. 10.1016/0012-365X(86)90024-5, Discrete Math. 61 (1986), 21–26. (1986) MR0850926DOI10.1016/0012-365X(86)90024-5
  3. 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) (2001) MR1962414
  4. The GAP Group, Gap-groups, algorithms, and programming, version 4.4, 2005, (http://www.gap-system.org), . 
  5. 10.1016/0012-365X(94)00250-M, Discrete Math. 148 (1996), 317–324. (1996) MR1368298DOI10.1016/0012-365X(94)00250-M
  6. 10.1023/B:CMAJ.0000042385.93571.58, Czechoslovak Math. J. 54 (2004), 465–485. (2004) MR2059267DOI10.1023/B:CMAJ.0000042385.93571.58
  7. A discrete iteration in number theory, BDTF Tud. Közl. 8 (1992), 71–91. (1992) Zbl0801.11011
  8. On the iteration of certain quadratic maps over GF ( p ) , Discrete Math. 277 (2004), 219–240. (2004) MR2033734

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