Symmetric Digraphs from Powers Modulo *n*

ABSTRACT

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*)

For each pair of positive integers

Cite this paper

nullG. 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.

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

References

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[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.

[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.