Ayman Badawi

Let $H$ be a finite abelian (commutative) group of order $n \geq 2$, and $m >1$ be an integer. We define the $m$-graph of $H$, denoted by $m-G(H)$, as a simple undirected graph with vertex set $H$, and two distinct vertices, $a, b \in H$, are connected by an edge if and only if $a^m = b$ or $b^m = a$. Several results regarding the properties of the $m$-$G(H)$ have been established.

Let $R$ be a finite product of integral domains and $D$ be a union of prime ideals (it is possible that $R$ is just an integral domain). Let $n \geq 1$ be a positive integer. This paper introduces the $n$-total graph of a $(R, D)$. The $n$-total graph of $(R, D)$, denoted by $n-T(R)$, is an undirected simple graph with vertex set $R$, such that two vertices $x, y$ in $R$ are connected by an edge if $x^n + y^n \in D$. In this paper, we study some graph properties and theoretical ring structure.

Let $R$ be a commutative ring with $1\not = 0$, $Z(R)$ be the set of all zero-divisors of $R$, and $n \geq 1$. This paper introduces the $n$-total graph of a commutative ring $R$. The $n$-total graph of a commutative ring $R$, denoted by $n-T(R)$, is an undirected simple graph with vertex set $R$, such that two vertices $x, y$ in $R$ are connected by an edge if $x^n + y^n$ in $Z(R)$. Note that if $n =1$, then the $1$-total graph of $R$ is the total graph of $R$ in the sense of Anderson-Badawi's paper on the total graph of a commutative ring. In this paper, we study some graph properties and theoretical ring structure.

Let S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph $\Gamma (S)$ with vertices $Z{(S)}^{*}=Z(S)\setminus \left\{0\right\},$ and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we introduce and study the n-zero-divisor graph of S as the (simple) graph ${\Gamma }_n(S)$ with vertices $Z_n{(S)}^{*}=\left\{x^n\right|x\in Z(S)\left\}\setminus \right\{0\},$ and distinct vertices x and y are adjacent if and only if xy = 0. Thus each ${\Gamma }_n(S)$ is an induced subgraph of $\Gamma (S)={\Gamma }_1(S).$ We pay particular attention to $\mathit{diam}({\Gamma }_n(S)),gr({\Gamma }_n(S)),$ and the case when S is a commutative ring with $1\ne 0.$ We also consider several other types of “n-zero-divisor” graphs and commutative rings such that some power of every element (or zero-divisor) is idempotent.

For a partially ordered set $(A, \le)$, let $G_A$ be the simple, undirected graph with vertex set $A$ such that two vertices $a \neq b\in A$ are adjacent if either $a \le b$ or $b \le a$. We call $G_A$ the \emph{partial order graph} or \emph{comparability graph} of $A$. Further, we say that a graph $G$ is a partial order graph if there exists a partially ordered set $A$ such that $G = G_A$. For a class $\mathcal{C}$ of simple, undirected graphs and $n$, $m \ge 1$, we define the Ramsey number $\mathcal{R}_{\mathcal{C}}(m,n)$ with respect to $\mathcal{C}$ to be the minimal number of vertices $r$ such that every induced subgraph of an arbitrary partial order graph consisting of $r$ vertices contains either a complete $n$-clique $K_n$ or an independent set consisting of $m$ vertices. In this paper, we determine the Ramsey number with respect to some classes of partial order graphs. Furthermore, some implications of Ramsey numbers in ring theory are discussed.

Let R be a commutative ring with $1\cancel{=}0$ and n a positive integer. A proper ideal I of R is an n-semiprimary ideal of R if whenever $x^n{y}^n\in I$ for $x,y\in R,$ then $x^n\in I$ or $y^n\in I.$ Let R be an integral domain with quotient field K. A proper ideal I of R is an n-powerful ideal of R if whenever $x^n{y}^n\in I$ for $x,y\in K,$ then $x^n\in R$ or $y^n\in R;$ and I is an n-powerful semiprimary ideal of R if whenever $x^n{y}^n\in I$ for $x,y\in K,$ then $x^n\in I$ or $y^n\in I.$ If every prime ideal of R is an n-powerful semiprimary ideal of R, then R is an n-pseudo-valuation domain (n-PVD). In this paper, we study the above concepts and relate them to several generalizations of pseudo-valuation domains.