Cut Vertices in Zero-Divisor Graphs of Finite Commutative Rings

A cut vertex of a connected graph is a vertex whose removal would result in a graph having two or more connected components. We examine the presence of cut vertices in zero-divisor graphs of finite commutative rings and provide a partial classification of the rings in which they appear.     

Classification of Rings with Genus One Zero-Divisor Graphs

This paper investigates properties of the zero-divisor graph of a commutative ring and its genus. In particular, we determine all isomorphism classes of finite commutative rings with identity whose zero-divisor graph has genus one.     

Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings

For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ? ?₂ × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.     

Zero-Divisor Graphs of Matrices Over Commutative Rings

We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a commutative ring R and that of the matrix ring M _n (R).     

Infinite Rings with Planar Zero-Divisor Graphs

In this article, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of ?₂-graded algebras. Essentially, the results are the same as in the classical case, except that the notion of a ?₂-graded division algebra needs to be modified. We classify all finite dimensional ?₂-graded division algebras over ? and ?.     

Cut Vertices and Degree One Vertices of Zero-Divisor Graphs

This article continues to examine cut vertices in the zero-divisor graphs of commutative rings with 1. The main result is that, with only seven known exceptions, the zero-divisor graph of a commutative ring has a cut vertex if and only if the graph has a degree one vertex. This naturally leads to an examination of the degree one vertices of zero-divisor graphs.     

Zero-Divisor Graphs of Polynomials and Power Series Over Commutative Rings

ABSTRACT

We recall several results about zero-divisor graphs of commutative rings. Then we examine the preservation of diameter and girth of the zero-divisor graph under extension to polynomial and power series rings.     

The Total Graphs of Finite Rings

In this paper we extend the study of total graphs τ(R) to noncommutative finite rings R. We prove that τ(R) is connected if and only if R is not local, and we see that in that case τ(R) is always Hamiltonian. We also find an upper bound for the domination number of τ(R) for all finite rings R.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.     

Dominating Sets of Some Graphs Associated to Commutative Rings

Let G = (V, E) be a graph. A set S ? V is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this article is to study and characterize the dominating sets of the zero-divisor graph Γ(R) and ideal-based zero-divisor graph Γ _I (R) of a commutative ring R.     

The Isomorphism Problem for Integral Group Rings of a Complete Monomial Group

Let G be a complete monomial group with abelian base, namely, G = AwrSym _m , the wreath product of a finite abelian group A with the symmetric group on m letters. Then the group G is determined by its integral group ring.     

The Zero-Divisor Graph Associated to a Semigroup

The zero-divisor graph of a commutative semigroup with zero is the graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices adjacent if the product of the corresponding elements is zero. New criteria to identify zero-divisor graphs are derived using both graph-theoretic and algebraic methods. We find the lowest bound on the number of edges necessary to guarantee a graph is a zero-divisor graph. In addition, the removal or addition of vertices to a zero-divisor graph is investigated by using equivalence relations and quotient sets. We also prove necessary and sufficient conditions for determining when regular graphs and complete graphs with more than two triangles attached are zero-divisor graphs. Lastly, we classify several graph structures that satisfy all known necessary conditions but are not zero-divisor graphs.     

On Realizing Zero-Divisor Graphs

An algorithm is presented for constructing the zero-divisor graph of a direct product of integral domains. Moreover, graphs which are realizable as zero-divisor graphs of direct products of integral domains are classified, as well as those of Boolean rings. In particular, graphs which are realizable as zero-divisor graphs of finite reduced commutative rings are classified.     

Zero-divisor Labelings of Graphs

This paper introduces the notions of a zero-divisor labeling and the zero-divisor index of a graph using the zero-divisors of a commutative ring. Viewed in this way, the usual zero-divisor graph is a maximal graph with respect to a zero-divisor labeling. We also study optimal zero-divisor labelings of a finite graph.     

Smarandache Vertices of the Graphs Associated to the Commutative Rings

Let R be a commutative ring with identity 1 ≠ 0. A nonzero element a in R is said to be a Smarandache zero-divisor if there exist three different nonzero elements x, y, and b (≠ a) in R such that ax = ab = by = 0, but xy ≠ 0. We will generalize this notion to the Smarandache vertex of an arbitrary simple graph and characterize the Smarandache zero-divisors of commutative rings (resp. with respect to an ideal) via their associated zero-divisor graphs. We illustrate them with examples and prove some interesting results about them.     

Zero-Divisor Graph with Respect to an Ideal

ABSTRACT

Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by Γ _I (R), is the graph whose vertices are the set {x ? R\I | xy ? I for some y ? R\I} with distinct vertices x and y adjacent if and only if xy ? I. In the case I = 0, Γ₀(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between Γ _I (R) ? Γ _J (S) and Γ(R/I) ? Γ(S/J). We also discuss when Γ _I (R) is bipartite. Finally we give some results on the subgraphs and the parameters of Γ _I (R).     

On Commuting Graphs of Finite Matrix Rings

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M _n (F)), for a finite field F and n ≥ 2, then R ≈ M _n (F). Here we prove this conjecture when n = 2.     

Commuting Graphs of Group Algebras

The commuting graph of a ring R, denoted by Γ(R), is a graph of all whose vertices are noncentral elements of R, and 2 distinct vertices x and y are adjacent if and only if xy = yx. In this article we investigate some graph-theoretic properties of Γ(kG), where G is a finite group, k is a field, and 0 ≠ |G| ∈k. Among other results it is shown that if G is a finite nonabelian group and k is an algebraically closed field, then Γ(kG) is not connected if and only if |G| = 6 or 8. For an arbitrary field k, we prove that Γ(kG) is connected if G is a nonabelian finite simple group or G′ ≠ G″ and G″ ≠ 1.