On iteration digraph and zero-divisor graph of the ring ${\mathbb{Z}}_n$

Ju, Tengxia, and Wu, Meiyun. "On iteration digraph and zero-divisor graph of the ring $\mathbb {Z}_n$." Czechoslovak Mathematical Journal 64.3 (2014): 611-628. <http://eudml.org/doc/262167>.

@article{Ju2014,
abstract = {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=\lbrace 0,1,\cdots ,n-1\rbrace $ 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 $a^5\equiv b\hspace\{4.44443pt\}(\@mod \; n)$. 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. 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.},
author = {Ju, Tengxia, Wu, Meiyun},
journal = {Czechoslovak Mathematical Journal},
keywords = {iteration digraph; zero-divisor graph; tree; cycle; vertex-connectivity; iteration digraph; zero-divisor graph; tree; cycle; vertex-connectivity},
language = {eng},
number = {3},
pages = {611-628},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On iteration digraph and zero-divisor graph of the ring $\mathbb \{Z\}_n$},
url = {http://eudml.org/doc/262167},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Ju, Tengxia
AU - Wu, Meiyun
TI - On iteration digraph and zero-divisor graph of the ring $\mathbb {Z}_n$
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 3
SP - 611
EP - 628
AB - 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=\lbrace 0,1,\cdots ,n-1\rbrace $ 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 $a^5\equiv b\hspace{4.44443pt}(\@mod \; n)$. 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. 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.
LA - eng
KW - iteration digraph; zero-divisor graph; tree; cycle; vertex-connectivity; iteration digraph; zero-divisor graph; tree; cycle; vertex-connectivity
UR - http://eudml.org/doc/262167
ER -