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.     

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 bipartite zero-divisor graphs

A (finite or infinite) complete bipartite graph together with some end vertices all adjacent to a common vertex is called a complete bipartite graph with a horn. For any bipartite graph G, we show that G is the graph of a commutative semigroup with 0 if and only if it is one of the following graphs: star graph, two-star graph, complete bipartite graph, complete bipartite graph with a horn. We also prove that a zero-divisor graph is bipartite if and only if it contains no triangles. In addition, we give all corresponding zero-divisor semigroups of a class of complete bipartite graphs with a horn and determine which complete r-partite graphs with a horn have a corresponding semigroup for r≥3.     

Zero-Divisor Graphs for Group Rings

Let R be a commutative ring with 1 ≠ 0, G be a nontrivial finite group, and let Z(R) be the set of zero divisors of R. The zero-divisor graph of R is defined as the graph Γ(R) whose vertex set is Z(R)* = Z(R)?{0} and two distinct vertices a and b are adjacent if and only if ab = 0. In this paper, we investigate the interplay between the ring-theoretic properties of group rings RG and the graph-theoretic properties of Γ(RG). We characterize finite commutative group rings RG for which either diam(Γ(RG)) ≤2 or gr(Γ(RG)) ≥4. Also, we investigate the isomorphism problem for zero-divisor graphs of group rings. First, we show that the rank and the cardinality of a finite abelian p-group are determined by the zero-divisor graph of its modular group ring. With the notion of zero-divisor graphs extended to noncommutative rings, it is also shown that two finite semisimple group rings are isomorphic if and only if their zero-divisor graphs are isomorphic. Finally, we show that finite noncommutative reversible group rings are determined by their zero-divisor graphs.     

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.     

Weakly central-vertex complete graphs with applications to commutative rings

The zero-divisor graph of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy=0. In this paper, a decomposition theorem is provided to describe weakly central-vertex complete graphs of radius 1. This characterization is then applied to the class of zero-divisor graphs of commutative rings. For finite commutative rings whose zero-divisor graphs are not isomorphic to that of Z₄[X]/(X²), it is shown that weak central-vertex completeness is equivalent to the annihilator condition. Furthermore, a schema for describing zero-divisor graphs of radius 1 is provided.     

Zero-Divisor Semigroups and Some Simple Graphs

In this article, we study commutative zero-divisor semigroups determined by graphs. We prove that for all n ≥ 4, the complete graph K _n together with two end vertices has a unique corresponding zero-divisor semigroup, while the complete graph K _n together with three end vertices has no corresponding semigroups. We determine all the twenty zero-divisor semigroups whose zero-divisor graphs are the complete graph K ₃ together with an end vertex.     

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.     

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.     

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 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.     

Corrigendum to “On zero-divisor graphs of small finite commutative rings”: [Discrete Math. 307 (2007) 1155-1166]

This note provides some minor corrections to the article “On zero-divisor graphs of small finite commutative rings”.     

Some Remarks on the Graph Γ I (R)

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)?{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ _I (R), with vertices {x ∈ R?I | xy ∈ I for some y ∈ R?I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ _I (R). We also determine when Γ _I (R) is either a complete graph or a complete bipartite graph and investigate when Γ _I (R) ? Γ(S) for some commutative ring S.     

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).     

Generalized Irreducible Divisor Graphs

In 1988, Beck introduced the notion of a zero-divisor graph of a commutative rings with 1. There have been several generalizations in recent years. In particular, in 2007 Coykendall and Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially τ-factorization. The goal of this paper is to synthesize the notions of τ-factorization and irreducible divisor graphs in domains. We will define a τ-irreducible divisor graph for nonzero non unit elements of a domain. We show that, by studying τ-irreducible divisor graphs, we find equivalent characterizations of several finite τ-factorization properties.     

Planar Hypohamiltonian Graphs on 40 Vertices

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer‐aided generation of certain families of planar graphs with girth 4 and a fixed number of 4‐faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamiltonian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypotraceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest planar hypohamiltonian graph of girth 5 has 45 vertices.     

上连通和超连通三次点传递图(英)

图G称为上连通的,若对每个最小割集C,G-C有孤立点.G称为超连通的,若对每个最小割集G,G-C恰有两个连通分支,且其中之一为孤立点.本文刻划了上连通和超连通三次点传递图.     

Spanning Trees with a Bounded Number of Branch Vertices in a Claw-Free Graph

Let k be a non-negative integer. A branch vertex of a tree is a vertex of degree at least three. We show two sufficient conditions for a connected claw-free graph to have a spanning tree with a bounded number of branch vertices: (i) A connected claw-free graph has a spanning tree with at most k branch vertices if its independence number is at most 2k + 2. (ii) A connected claw-free graph of order n has a spanning tree with at most one branch vertex if the degree sum of any five independent vertices is at least n ? 2. These conditions are best possible. A related conjecture also is proposed.