Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The new extension the zero-divisor graph $$\widetilde{\Gamma }(R)$$ vertices $$Z(R)^{\star }$$ , two distinct x y are adjacent if only $$xy=0$$ or $$x+y\in Z(R)$$ . In this article, we study, in general case, For any R, provide sufficient necessary conditions for }(R[x_1,\dots x_n])$$ to complete. At last, pres...

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertices in W ∗(R), which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in W ∗(R) are adjacent if and only if a / ∈ bR and b / ∈ aR. In this paper, we characterize all commutative rings whose cozero-divisor graphs are forest, star, doubl...

In this paper, we introduce the multiplicative zero-divisor graph of a multiplicative lattice and study Beck-like coloring of such graphs. Further, it is proved that for such graphs, the chromatic number and the clique number need not be equal. On the other hand, if a multiplicative lattice L is reduced, then the chromatic number and the clique number of the multiplicative zero-divisor graph of...

The aim of this article to follow the properties zero-divisor graph special idealization ring. We study wiener index zero-divisors some ring R(+)M and find clique number Γ(R(+)M) is ω (Γ(R(+)M)) = |M| − 1, where R an integral domain. also discuss when are Hamiltonian graph.

L-zero-divisor graphs of L-commutative rings have been introduced and studied in [5]. Here we consider L-zero-divisor graphs of a finite direct product of L-commutative rings. Specifically, we look at the preservation, or lack thereof, of the diameter and girth of the L-zirodivisor graph of a L-ring when extending to a finite direct product of L-commutative rings.

Let S be a semiring and let Z(S)∗ be its set of nonzero zero divisors. We denote the zero divisor graph of S by Γ(S) whose vertex set is Z(S)∗ and there is an edge between the vertices x and y (x 6= y) in Γ(S) if and only if either xy = 0 or yx = 0. In this paper we study the zero divisor graph of the semiring of matrices Mn(B), (n > 1) over the Boolean semiring B. We investigate the properties...