Journal of Applied Mathematics

On Simple Graphs Arising from Exponential Congruences

M. Aslam Malik and M. Khalid Mahmood

Full-text: Open access

Abstract

We introduce and investigate a new class of graphs arrived from exponential congruences. For each pair of positive integers a and b , let G ( n ) denote the graph for which V = { 0 , 1 , , n 1 } is the set of vertices and there is an edge between a and b if the congruence a x b  ( mod  n ) is solvable. Let n = p 1 k 1 p 2 k 2 p r k r be the prime power factorization of an integer n , where p 1 < p 2 < < p r are distinct primes. The number of nontrivial self-loops of the graph G ( n ) has been determined and shown to be equal to i = 1 r ( ϕ ( p i k i ) + 1 ) . It is shown that the graph G ( n ) has 2 r components. Further, it is proved that the component Γ p of the simple graph G ( p 2 ) is a tree with root at zero, and if n is a Fermat's prime, then the component Γ ϕ ( n ) of the simple graph G ( n ) is complete.

Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 292895, 10 pages.

Dates
First available in Project Euclid: 2 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.jam/1357153519

Digital Object Identifier
doi:10.1155/2012/292895

Mathematical Reviews number (MathSciNet)
MR2979465

Zentralblatt MATH identifier
1279.05074

Citation

Malik, M. Aslam; Mahmood, M. Khalid. On Simple Graphs Arising from Exponential Congruences. J. Appl. Math. 2012 (2012), Article ID 292895, 10 pages. doi:10.1155/2012/292895. https://projecteuclid.org/euclid.jam/1357153519


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