# On semiregular digraphs of the congruence ${x}^{k}\equiv y\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}n)$

Commentationes Mathematicae Universitatis Carolinae (2007)

- Volume: 48, Issue: 1, page 41-58
- ISSN: 0010-2628

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topSomer, Lawrence, and Křížek, Michal. "On semiregular digraphs of the congruence $x^k\equiv y \hspace{4.44443pt}(\@mod \; n)$." Commentationes Mathematicae Universitatis Carolinae 48.1 (2007): 41-58. <http://eudml.org/doc/250213>.

@article{Somer2007,

abstract = {We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ whose set of vertices is $H=\lbrace 0,1,\dots ,n-1\rbrace $ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b\hspace\{4.44443pt\}(\@mod \; n)$. The digraph $G(n,k)$ is semiregular if there exists a positive integer $d$ such that each vertex of the digraph has indegree $d$ or 0. Generalizing earlier results of the authors for the case in which $k=2$, we characterize all semiregular digraphs $G(n,k)$ when $k\ge 2$ is arbitrary.},

author = {Somer, Lawrence, Křížek, Michal},

journal = {Commentationes Mathematicae Universitatis Carolinae},

keywords = {Chinese remainder theorem; congruence; group theory; dynamical system; regular and semiregular digraphs; Chinese remainder theorem; congruence; group theory; dynamical system; regular and semiregular digraphs},

language = {eng},

number = {1},

pages = {41-58},

publisher = {Charles University in Prague, Faculty of Mathematics and Physics},

title = {On semiregular digraphs of the congruence $x^k\equiv y \hspace\{4.44443pt\}(\@mod \; n)$},

url = {http://eudml.org/doc/250213},

volume = {48},

year = {2007},

}

TY - JOUR

AU - Somer, Lawrence

AU - Křížek, Michal

TI - On semiregular digraphs of the congruence $x^k\equiv y \hspace{4.44443pt}(\@mod \; n)$

JO - Commentationes Mathematicae Universitatis Carolinae

PY - 2007

PB - Charles University in Prague, Faculty of Mathematics and Physics

VL - 48

IS - 1

SP - 41

EP - 58

AB - We assign to each pair of positive integers $n$ and $k\ge 2$ a digraph $G(n,k)$ whose set of vertices is $H=\lbrace 0,1,\dots ,n-1\rbrace $ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^k\equiv b\hspace{4.44443pt}(\@mod \; n)$. The digraph $G(n,k)$ is semiregular if there exists a positive integer $d$ such that each vertex of the digraph has indegree $d$ or 0. Generalizing earlier results of the authors for the case in which $k=2$, we characterize all semiregular digraphs $G(n,k)$ when $k\ge 2$ is arbitrary.

LA - eng

KW - Chinese remainder theorem; congruence; group theory; dynamical system; regular and semiregular digraphs; Chinese remainder theorem; congruence; group theory; dynamical system; regular and semiregular digraphs

UR - http://eudml.org/doc/250213

ER -

## References

top- Křížek M., Luca F., Somer L., 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS Books in Mathematics, vol. 9, Springer New York (2001). (2001) Zbl1010.11002MR1866957
- Lucheta C., Miller E., Reiter C., Digraphs from powers modulo $p$, Fibonacci Quart. (1996), 34 226-239. (1996) Zbl0855.05067MR1390409
- Niven I., Zuckerman H.S., Montgomery H.L., An Introduction to the Theory of Numbers, fifth edition, John Wiley & Sons, New York (1991). (1991) Zbl0742.11001MR1083765
- Somer L., Křížek M., On a connection of number theory with graph theory, Czechoslovak Math. J. 54 (2004), 465-485. (2004) Zbl1080.11004MR2059267
- Somer L., Křížek M., Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174-2185. (2006) Zbl1161.05323MR2255611
- Wilson B., Power digraphs modulo $n$, Fibonacci Quart. (1998), 36 229-239. (1998) Zbl0936.05049MR1627384

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