Previous |  Up |  Next


iteration digraph; height; Carmichael lambda function; fixed point; regular digraph
A power digraph, denoted by $G(n,k)$, is a directed graph with $\mathbb Z_{n}=\{0,1,\dots ,n-1\}$ as the set of vertices and $E=\{(a,b)\colon a^{k}\equiv b\pmod n\}$ as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of $G(n,k)$ for $n\geq 1$ and $k\geq 2$ are determined. We also find an expression for the number of vertices at a specific height. Finally, we obtain necessary and sufficient conditions on $n$ such that each vertex of indegree $0$ of a certain subdigraph of $G(n,k)$ is at height $q\geq 1$.
[1] Burton, D. M.: Elementary Number Theory. McGraw-Hill (2007).
[2] Carlip, W., Mincheva, M.: Symmetry of iteration graphs. Czech. Math. J. 58 (2008), 131-145. DOI 10.1007/s10587-008-0009-8 | MR 2402530 | Zbl 1174.05048
[3] Carmichael, R. D.: Note on a new number theory function. Am. Math. Soc. Bull. 16 (1910), 232-238. DOI 10.1090/S0002-9904-1910-01892-9 | MR 1558896
[4] Chartrand, G., Oellermann, O. R.: Applied and Algorithmic Graph Theory. International Series in Pure and Applied Mathematics McGraw-Hill (1993) \MR 1211413. MR 1211413
[5] Deo, N.: Graph theory with Application to Engineering and Computer Sciences. Prentice-Hall Series in Automatic Computation. Englewood Cliffs, N.J.: Prentice-Hall (1974). MR 0360322
[6] Ellson, J., Gansner, E., Koutsofios, L., North, S. C., Woodhull, G.: Graphviz--open source graph drawing tools. Mutzel, Petra (ed.) et al., Graph drawing. 9th international symposium, GD 2001, Vienna, Austria, September 23-26, 2001 Revised papers. Berlin: Springer. Lect. Notes Comput. Sci. 2265 (2002), 483-484. MR 1962414 | Zbl 1054.68583
[7] Husnine, S. M., Ahmad, U., Somer, L.: On symmetries of power digraphs. Util. Math. 85 (2011), 257-271. MR 2840802
[8] Kramer-Miller, J.: Structural properties of power digraphs modulo $n$. Proceedings of the 2009 Midstates Conference on Undergraduate Research in Computer Science and Mathematics, Oberlin, Ohio (2009), 40-49.
[9] Lucheta, C., Miller, E., Reiter, C.: Digraphs from powers modulo $p$. Fibonacci Q. 34 (1996), 226-239. MR 1390409 | Zbl 0855.05067
[10] Somer, L., Křížek, M.: On a connection of number theory with graph theory. Czech. Math. J. 54 (2004), 465-485. DOI 10.1023/B:CMAJ.0000042385.93571.58 | MR 2059267
[11] Somer, L., Křížek, M.: Structure of digraphs associated with quadratic congruences with composite moduli. Discrete Math. 306 (2006), 2174-2185. DOI 10.1016/j.disc.2005.12.026 | MR 2255611
[12] Somer, L., Křížek, M.: On semiregular digraphs of the congruence $x^{k}\equiv y \pmod n$. Commentat. Math. Univ. Carol. 48 (2007), 41-58. MR 2338828
[13] Somer, L., Křížek, M.: On symmetric digraphs of the congruence $x^{k}\equiv y \pmod n$. Discrete Math. 309 (2009), 1999-2009. DOI 10.1016/j.disc.2008.04.009 | MR 2510326
[14] Wilson, B.: Power digraphs modulo $n$. Fibonacci Q. 36 (1998), 229-239. MR 1627384 | Zbl 0936.05049
[15] MATLAB, The language of technical computing (version (R14)).
Partner of
EuDML logo