# Article

Keywords:
iteration digraph; zero-divisor graph; tree; cycle; vertex-connectivity
Summary:
In the first part, we assign to each positive integer $n$ a digraph $\Gamma (n,5),$ whose set of vertices consists of elements of the ring $\mathbb {Z}_n=\{0,1,\cdots ,n-1\}$ with the addition and the multiplication operations modulo $n,$ and for which there is a directed edge from $a$ to $b$ if and only if $a^5\equiv b\pmod n$. Associated with $\Gamma (n,5)$ are two disjoint subdigraphs: $\Gamma _1(n,5)$ and $\Gamma _2(n,5)$ whose union is $\Gamma (n,5).$ The vertices of $\Gamma _1(n,5)$ are coprime to $n,$ and the vertices of $\Gamma _2(n,5)$ are not coprime to $n.$ In this part, we study the structure of $\Gamma (n,5)$ in detail. \endgraf In the second part, we investigate the zero-divisor graph $G(\mathbb {Z}_n)$ of the ring $\mathbb {Z}_n.$ Its vertex- and edge-connectivity are discussed.
References:
 Akbari, S., Mohammadian, A.: On the zero-divisor graph of a commutative ring. J. Algebra 274 (2004), 847-855. DOI 10.1016/S0021-8693(03)00435-6 | MR 2043378 | Zbl 1085.13011
 Akhtar, R., Lee, L.: Connectivity of the zero-divisor graph of finite rings. https//www.researchgate.net/publication/228569713
 Beck, I.: Coloring of commutative rings. J. Algebra 116 (1988), 208-226. DOI 10.1016/0021-8693(88)90202-5 | MR 0944156 | Zbl 0654.13001
 Carmichael, R. D.: Note on a new number theory function. Amer. Math. Soc. Bull. (2) 16 (1910), 232-238. DOI 10.1090/S0002-9904-1910-01892-9 | MR 1558896
 Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory. 2nd Graduate Texts in Mathematics 84 Springer, New York (1990). MR 1070716 | Zbl 0712.11001
 Křížek, M., Luca, F., Somer, L.: 17 Lectures on Fermat Numbers. From Number Theory to Geometry. CMS Books in Mathematics 9 Springer, New York (2001). MR 1866957 | Zbl 1010.11002
 Skowronek-Kaziów, J.: Some digraphs arising from number theory and remarks on the zero-divisor graph of the ring $\mathbb{Z}_n$. Inf. Process. Lett. 108 (2008), 165-169. DOI 10.1016/j.ipl.2008.05.002 | MR 2452147
 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
 Somer, L., Křížek, M.: Stucture 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
 Somer, L., Křížek, M.: The structure of digraphs associated with the congruence $x^k\equiv y\pmod n$. Czech. Math. J. 61 (2011), 337-358. DOI 10.1007/s10587-011-0079-x | MR 2905408 | Zbl 1249.11006
 Vasiga, T., Shallit, J.: On the iteration of certain quadratic maps over GF($p$). Discrete Math. 277 (2004), 219-240. DOI 10.1016/S0012-365X(03)00158-4 | MR 2033734 | Zbl 1045.11086
 Wilson, B.: Power digraphs modulo $n$. Fibonacci Q. 36 (1998), 229-239. MR 1627384 | Zbl 0936.05049