Characterization of power digraphs modulo n

Uzma Ahmad; Syed Husnine

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 3, page 359-367
  • ISSN: 0010-2628

Abstract

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A power digraph modulo n , denoted by G ( n , k ) , is a directed graph with Z n = { 0 , 1 , ⋯ , n - 1 } as the set of vertices and E = { ( a , b ) : a k ≡ b ( mod n ) } as the edge set, where n and k are any positive integers. In this paper we find necessary and sufficient conditions on n and k such that the digraph G ( n , k ) has at least one isolated fixed point. We also establish necessary and sufficient conditions on n and k such that the digraph G ( n , k ) contains exactly two components. The primality of Fermat number is also discussed.

How to cite

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Ahmad, Uzma, and Husnine, Syed. "Characterization of power digraphs modulo $n$." Commentationes Mathematicae Universitatis Carolinae 52.3 (2011): 359-367. <http://eudml.org/doc/246164>.

@article{Ahmad2011,
abstract = {A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with $Z_\{n\}=\lbrace 0,1,\dots , n-1\rbrace $ as the set of vertices and $E=\lbrace (a,b): a^\{k\}\equiv b\hspace\{4.44443pt\}(\@mod \; n)\rbrace $ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ contains exactly two components. The primality of Fermat number is also discussed.},
author = {Ahmad, Uzma, Husnine, Syed},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {iteration digraph; isolated fixed points; Charmichael lambda function; Fermat numbers; Regular digraphs; iteration digraph; isolated fixed point; Fermat number; regular digraph},
language = {eng},
number = {3},
pages = {359-367},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Characterization of power digraphs modulo $n$},
url = {http://eudml.org/doc/246164},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Ahmad, Uzma
AU - Husnine, Syed
TI - Characterization of power digraphs modulo $n$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 3
SP - 359
EP - 367
AB - A power digraph modulo $n$, denoted by $G(n,k)$, is a directed graph with $Z_{n}=\lbrace 0,1,\dots , n-1\rbrace $ as the set of vertices and $E=\lbrace (a,b): a^{k}\equiv b\hspace{4.44443pt}(\@mod \; n)\rbrace $ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G(n,k)$ contains exactly two components. The primality of Fermat number is also discussed.
LA - eng
KW - iteration digraph; isolated fixed points; Charmichael lambda function; Fermat numbers; Regular digraphs; iteration digraph; isolated fixed point; Fermat number; regular digraph
UR - http://eudml.org/doc/246164
ER -

References

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  15. Husnine S.M., Ahmad U., Somer L., On symmetries of power digraphs, Utilitas Mathematica(to appear). MR2840802
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