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.
Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann(xy) \neqann R(x)\cup annR(y), where for z \in R, annR(z) = {r \in R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n>1., Mojgan Afkhami, Kazem Khashyarmanesh, Zohreh Rajabi., and Obsahuje bibliografii