On the heights of power digraphs modulo $n$

Title:

On the heights of power digraphs modulo $n$

Creator:

Ahmad, Uzma and Syed, Husnine

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:805ce3fd-6928-4330-80b7-f22891cd5b5e
uuid:805ce3fd-6928-4330-80b7-f22891cd5b5e

Subject:

iteration digraph, height, Carmichael lambda function, fixed point, and regular digraph

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

A power digraph, denoted by $G(n,k)$, is a directed graph with $\mathbb Z_{n}=\{0,1,\dots ,n-1\}$ as the set of vertices and $E=\{(a,b)\colon a^{k}\equiv b\pmod n\}$ 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 $n\geq 1$ and $k\geq 2$ 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 $q\geq 1$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

541-556

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:2

Harvested from:

CDK

Metadata only:

false