# On the heights of power digraphs modulo $n$

Czechoslovak Mathematical Journal (2012)

- Volume: 62, Issue: 2, page 541-556
- ISSN: 0011-4642

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topAhmad, Uzma, and Syed, Husnine. "On the heights of power digraphs modulo $n$." Czechoslovak Mathematical Journal 62.2 (2012): 541-556. <http://eudml.org/doc/246239>.

@article{Ahmad2012,

abstract = {A power digraph, denoted by $G(n,k)$, is a directed graph with $\mathbb \{Z\}_\{n\}=\lbrace 0,1,\dots ,n-1\rbrace $ as the set of vertices and $E=\lbrace (a,b)\colon a^\{k\}\equiv b\hspace\{4.44443pt\}(\@mod \; n)\rbrace $ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\ge 1$ and $k\ge 2$ are determined. We also find an expression for the number of vertices at a specific height. Finally, we obtain necessary and sufficient conditions on $n$ such that each vertex of indegree $0$ of a certain subdigraph of $G(n,k)$ is at height $q\ge 1$.},

author = {Ahmad, Uzma, Syed, Husnine},

journal = {Czechoslovak Mathematical Journal},

keywords = {iteration digraph; height; Carmichael lambda function; fixed point; regular digraph; iteration digraph; height; Carmichael lambda function; fixed point; regular digraph},

language = {eng},

number = {2},

pages = {541-556},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {On the heights of power digraphs modulo $n$},

url = {http://eudml.org/doc/246239},

volume = {62},

year = {2012},

}

TY - JOUR

AU - Ahmad, Uzma

AU - Syed, Husnine

TI - On the heights of power digraphs modulo $n$

JO - Czechoslovak Mathematical Journal

PY - 2012

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 62

IS - 2

SP - 541

EP - 556

AB - A power digraph, denoted by $G(n,k)$, is a directed graph with $\mathbb {Z}_{n}=\lbrace 0,1,\dots ,n-1\rbrace $ as the set of vertices and $E=\lbrace (a,b)\colon a^{k}\equiv b\hspace{4.44443pt}(\@mod \; n)\rbrace $ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\ge 1$ and $k\ge 2$ are determined. We also find an expression for the number of vertices at a specific height. Finally, we obtain necessary and sufficient conditions on $n$ such that each vertex of indegree $0$ of a certain subdigraph of $G(n,k)$ is at height $q\ge 1$.

LA - eng

KW - iteration digraph; height; Carmichael lambda function; fixed point; regular digraph; iteration digraph; height; Carmichael lambda function; fixed point; regular digraph

UR - http://eudml.org/doc/246239

ER -

## References

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