Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 2 (2018), 181-194.

Finding cycles in the $k$-th power digraphs over the integers modulo a prime

Greg Dresden and Wenda Tu

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For p prime and k2, let us define Gp(k) to be the digraph whose set of vertices is {0,1,2,,p1} such that there is a directed edge from a vertex a to a vertex b if akb modp. We find a new way to decide if there is a cycle of a given length in a given graph Gp(k).

Article information

Involve, Volume 11, Number 2 (2018), 181-194.

Received: 21 January 2014
Revised: 8 June 2017
Accepted: 21 June 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 11R04: Algebraic numbers; rings of algebraic integers

digraphs cycles graph theory number theory


Dresden, Greg; Tu, Wenda. Finding cycles in the $k$-th power digraphs over the integers modulo a prime. Involve 11 (2018), no. 2, 181--194. doi:10.2140/involve.2018.11.181.

Export citation


  • A. S. Bang, “Taltheoretiske undersølgelser”, Tidsskrift f. Math. 4:5 (1886), 70–80, 130–137.
  • J. A. Gallian, Contemporary abstract algebra, 7th ed., Brooks/Cole, Belmont, CA, 2010.
  • C. Lucheta, E. Miller, and C. Reiter, “Digraphs from powers modulo $p$”, Fibonacci Quart. 34:3 (1996), 226–239.
  • I. Niven, H. S. Zuckerman, and H. L. Montgomery, An introduction to the theory of numbers, 5th ed., Wiley, New York, 1991.
  • M. Roitman, “On Zsigmondy primes”, Proc. Amer. Math. Soc. 125:7 (1997), 1913–1919.
  • L. Somer and M. Křížek, “On a connection of number theory with graph theory”, Czechoslovak Math. J. 54(129):2 (2004), 465–485.
  • L. Somer and M. Křížek, “On symmetric digraphs of the congruence $x^k\equiv y\pmod n$”, Discrete Math. 309:8 (2009), 1999–2009.
  • B. Wilson, “Power digraphs modulo $n$”, Fibonacci Quart. 36:3 (1998), 229–239.
  • K. Zsigmondy, “Zur Theorie der Potenzreste”, Monatsh. Math. Phys. 3:1 (1892), 265–284.