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2012 On Simple Graphs Arising from Exponential Congruences
M. Aslam Malik, M. Khalid Mahmood
J. Appl. Math. 2012: 1-10 (2012). DOI: 10.1155/2012/292895

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.

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M. Aslam Malik. M. Khalid Mahmood. "On Simple Graphs Arising from Exponential Congruences." J. Appl. Math. 2012 1 - 10, 2012. https://doi.org/10.1155/2012/292895

Information

Published: 2012
First available in Project Euclid: 2 January 2013

zbMATH: 1279.05074
MathSciNet: MR2979465
Digital Object Identifier: 10.1155/2012/292895

Rights: Copyright © 2012 Hindawi

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