The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative rin...

We recall several results of zero divisor graphs of commutative rings. We then examine the preservation of the diameter of the zero divisor graph of polynomial and power series rings.

Consider the (p,q) simple connected graph . The sum absolute values of spectrum quotient matrix a make up graph's energy. objective this study is to examine energy identity graphs and zero-divisor commutative rings using group theory, applications. In study, derived from few classes ring R are examined.

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in   R     W R  , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and   W R  a bR  b aR  . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs    ...

We introduce the notion of semibreak divisors on metric graphs and prove that every effective divisor class (of degree at most genus) has a representative. This appropriately generalizes break (in equal to genus). provide an algorithm efficiently compute such representatives. Semibreak tool establish some basic properties loci inside Picard groups graphs. are pure-dimensional polyhedral sets. a...

Let $\mathcal C_{n}$ be the Catalan monoid on $X_{n}=\{1,\ldots ,n\}$ under its natural order. In this paper, we describe sets of left zero-divisors, right zero-divisors and two sided C_{n}$; their numbers. For $n \geq 4$, define an undirected graph $\Gamma(\mathcal C_{n})$ associated with whose vertices are excluding zero element $\theta$ distinct $\alpha$ $\beta$ joined by edge in case $\alph...