The power digraphs of safe primes

Document Type: Research Paper

Authors

1 Department of‎ ‎Mathematics‎, ‎University‎ ‎of the Punjab‎, ‎New Campus‎, ‎Lahore‎, ‎Pakistan.

2 Department of Humanities‎ ‎and Sciences‎, ‎National University‎ ‎of Computer and Emerging Sciences(FAST)‎, ‎Lahore Campus‎, ‎Lahore‎, ‎Pakistan.

Abstract

A power digraph, denoted by $G(n,k)$, is a directed graph with $Z_{n}={0,1,..., n-1}$ as the set of vertices and
$L={(x,y):x^{k}\equiv y~(mod , n)}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper, the structure of $G(2q+1,k)$, where $q$ is a Sophie Germain prime is investigated. The primality tests for the integers of the form $n=2q+1$ are established in terms of the structure of components of $G(n,k)$. The digraphs in which all components look like directed star graphs are completely classified. This work generalizes the results of M. Krizekek, L. Somer, Sophie Germain Little Suns, Math. Slovaca 54(5) (2004), 433-442.

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