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Symmetric Digraphs from Powers Modulo *n* ()

For each pair of positive integers

*n*and*k*, let G(n,k) denote the digraph whose set of vertices is*H*= {0,1,2,···,*n*– 1} and there is a directed edge from*a*∈*H*to*b*∈*H*if*a*≡*b*(mod n). The digraph*G*(*n*,*k*) is symmetric if its connected component can be partitioned into isomorphic pairs. In this paper we obtain all symmetric*G*(*n*,*k*)Keywords

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G. Deng and P. Yuan, "Symmetric Digraphs from Powers Modulo

*n*,"*Open Journal of Discrete Mathematics*, Vol. 1 No. 3, 2011, pp. 103-107. doi: 10.4236/ojdm.2011.13013.Conflicts of Interest

The authors declare no conflicts of interest.

[1] | W. Carlip and M. Mincheva, “Symmetry of Iteration Digraphs,” Czechoslovak Mathematic Journal, Vol. 58, No. 1, 2008, pp. 131-145. doi:10.1007/s10587-008-0009-8 |

[2] | G. Chartrand and L. Lesnidk, “Graphs and Digraphs (3rd Edition),” Chapman Hall, London, 1996. |

[3] | Wun-Seng Chou and Igor E. Shparlinski, “On the Cycle Structure of Repeated Exponentiation Modulo a Prime,” Journal of Number Theory, Vol.107, No. 2, 2004, pp. 345-356. doi:10.1016/j.jnt.2004.04.005 |

[4] | Joe Kramer-Miller, “Structural Properties of Power Digraphs Mudulo n,” Manuscript. |

[5] | M. Krizek, F. Lucas and L. Somer, “17 Lectures on the Femat Numbers, from Number Theory to Geometry,” Springer, New York, 2001. |

[6] | C. Lucheta, E. Miller and C. Reiter, “Digraphs from Powers Modulo p,” Fibonacci Quart, Vol. 34, 1996, pp. 226-239. |

[7] | I. Niven, H. S. Zuckerman and H. L. Montgomery, “An Introduction to the Theory of Numbers,” 5th Edition, John Wiley & Sons, New York, 1991. |

[8] | T. D. Rogers, “The Graph of the Square Mapping on the Prime Fields,” Discrete Mathematics, Vol. 148, No. 1-23, 1996, pp. 317-324. doi:10.1016/0012-365X(94)00250-M |

[9] | L. Somer and M. Krizek, “On a Connection of Number Theory with Graph Theory,” Czechoslovak Mathematic Journal, Vol. 54, No. 2, 2004, pp. 465-485. doi:10.1023/B:CMAJ.0000042385.93571.58 |

[10] | L. Somer and M. Krizek, “Structure of Digrphs Associated with Quadratic Congruences with Composite Moduli,” Discrete Mathematics, Vol. 306, No. 18, 2006, pp. 2174-2185. doi:10.1016/j.disc.2005.12.026 |

[11] | L. Somer and M. Krizek, “On Semiregular Digraphs of the Congruence xk ≡ y(mod n),” Commentationes Mathematicae Universitatis Carolinae, Vol. 48, No. 1, 2007, pp. 41-58. |

[12] | L. Szalay, “A Discrete Iteration in Number Theory,” BDTF Tud. KAozl, Vol. 8, 1992, pp. 71-91. |

[13] | L. Somer and M. Krizek, “On Symmetric Digrphs of the Congruence xk ≡ y (mod n),” Discrete Mathematics, Vol. 309, No. 8, 2009, pp. 1999-2009. doi:10.1016/j.disc.2008.04.009 |

[14] | B. Wilson, “Power Digraphs Modulo n,” Fibonacci Quart, Vol. 36, 1998, pp. 229-239. |

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