Maximum Matching of Zero-Divisor Graph Over a Commutative Ring

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

The Harary Index is an important topological parameter for examining the structure of a graph. This work presents a quantitative analysis of the structural features of zero-divisor graph using the Harary Index. In this research article, we compute the Harary Index for the zero-divisor graph of the ring of upper triangular matrices over a finite commutative ring and non commutative ring. In this study, we have computed the Harary index for directed and undirected zero-divisor graph obtained from upper triangular matrix [Formula: see text].

<abstract><p>The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $ R $ having a hyperideal $ I $, the $ I $-based zero-divisor graph $ \Gamma^{(I)}(R) $ associated with $ R $ is the simple graph whose vertices are the elements of $ R\setminus I $ having their hyperproduct in $ I $, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $ I $. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $ I $-based zero-divisor graph associated to the general hyperring $ \mathbb{Z}_n $ of the integers modulo $ n $, when $ n = 2p^mq $, with $ p $ and $ q $ two different odd primes, and fixing the hyperideal $ I $.</p></abstract>

Graphs serve as a bridge between algebra and numerous mathematical fields, including combinatorics, number theory, and computer science. Let Z(R) represent the set of zero-divisors in a commutative ring R. The zero-divisor graph of R, denoted by Γ(R), is an undirected graph with vertex set Z(R)∗ = Z(R) \\ {0}, where two vertices are adjacent if their product is zero. The Steiner tree and Steiner radial number are fundamental concepts in network optimization, offering cost-efficient strategies for connecting multiple points with minimal total distance. In this study, we compute the m-eccentricity of all vertices and determine the Steiner radial number of the zero-divisor graph of ℤ n . Additionally, for some positive integer m ≥ 2, we establish the existence of non-isomorphic zero-divisor graphs with Steiner radial number m. This work not only enhances the understanding of zero-divisor graphs but also provides a foundation for future studies in algebraic graph theory and network optimization.

Let [Formula: see text], [Formula: see text], [Formula: see text] denote the zero-divisor graph, compressed zero-divisor graph and annihilating ideal graph of a commutative ring [Formula: see text], respectively. In this paper, we prove that [Formula: see text] for a semisimple commutative ring [Formula: see text] and represent [Formula: see text] as a generalized join of a finite set of graphs. Further, we study the zero-divisor graph of a semisimple group-ring [Formula: see text] and proved several structural properties of [Formula: see text] and [Formula: see text], where [Formula: see text] is a field with [Formula: see text] elements and [Formula: see text] is a cyclic group with [Formula: see text] elements.

Approximating permanents and hafnians, Discrete Analysis 2017:2, 34 pp. The _permanent_ per$(A)$ of an $n\times n$ matrix $A$ is like the determinant but without the sign changes: that is, it is the sum $\sum_{\pi\in S_n}\prod_{i=1}^nA_{i\pi(i)}$. If $A$ is the bipartite adjacency matrix of a bipartite graph $G$, then per$(A)$ is the number of perfect matchings in $G$. Whereas row-operations allow one to compute the determinant of a matrix efficiently, computing the permanent is known, by a famous result of Valiant, to be hard (in the sense that if you could compute it efficiently, then you would be able to give efficient solutions to a wide variety of enumeration problems, and would in particular have shown that P=NP). However, another famous result, due to Jerrum, Sinclair and Vigoda, which built on earlier work of Jerrum and Sinclair, shows that there are probabilistic algorithms that run in polynomial time and approximate the permanent to within a factor $1+\epsilon$ with high probability. This paper concerns the question of how well the permanent can be approximated using a _deterministic_ algorithm. The main result of this paper is that if $\delta,\epsilon>0$ then for each $n\in\mathbb N$ there is a polynomial $p$ in $n^2$ variables of degree $O_\delta(\log n+\log(1/\epsilon))$ such that if $A$ is any $n\times n$ matrix with all its entries between $\delta$ and 1, then the permanent of $A$ can be approximated to within a factor $1+\epsilon$ by $\exp(p(A))$. Here, of course, $p(A)$ is $p$ applied to the entries of $A$. Furthermore, $p(A)$ can be computed in time $n^{O_\delta(\log n+\log(1/\epsilon))}$. As mentioned above, the permanent of a 01-matrix counts the number of perfect matchings in a bipartite graph. If instead one is interested in the number of perfect matchings in a graph with $2n$ vertices (that is, the number of sets of $n$ disjoint edges), then the corresponding quantity is called the _hafnian_. For a general $2n\times 2n$ matrix $A$, the hafnian haf$(A)$ is the sum over all partitions of $\{1,\dots,2n\}$ into $n$ sets $\{i_r,j_r\}$ of the product $\prod_{r=1}^nA_{i_rj_r}$. The author proves the same result for hafnians that he obtains for permanents. This is interesting, because it is not known how to approximate hafnians even with a randomized algorithm. The permanent and hafnian are polynomials of degree $n$ in the matrix entries, so these results say that on the class of matrices being considered (that is, real matrices with all entries between $\delta$ and 1), they can be approximated by polynomials of much lower degree. The proofs depend on results that say that the two polynomials cannot be zero on a complex matrix that is close to this class. Similar results are proved for complex matrices with all their entries close to 1. In this case the methods carry over to permanents of multidimensional tensors, but multidimensional generalizations appear to be harder to obtain for the results mentioned above, for reasons explained in the paper, and are not currently known.

The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo psqt for all distinct primes p, q and . We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph ΓE(R) are derived for .

Let R be a commutative ring with non-zero identity and Z^∗(R) be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices x, y ∈ Z^∗(R) are adjacent if and only if xy = 0. In this paper, the eigenvalues of Γ(Zn) for n = p^2q^2, where p and q are distinct primes, are investigated. Also, the girth, diameter, clique number and stability number of this graph are found.

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc (G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

This paper presents an efficiently covered property of the zero-divisor graph of L (+) N, with L a commutative ring with the identity element and N an L−module. we present a complete characterization of when Γ (L (+) N) is well covered in terms of the annihilator structure of N. Moreover, new concept is introduced its r−well covered graphs: as far as a well-covered graph has all maximal independent sets of size r. They can further prove that if L is an integral domain, ann(N) = 0 and if the graph Γ (L (+) N) is well covered in the sense of (|N|− 1) −well covered. All this goes to show that at least a meaningful link must exist between the algebraic structural condition of idealization rings and the corresponding graph-theoretic properties of their associated zero-divisor graphs: something quite different from the former studies on clique numbers and independence numbers in these structures.

The equivalence class [r] of an element r ∈ R is the set of zero-divisors s such that ann(r) = ann(s), that is, [r] = {s ∈ R : ann(r) = ann(s). The compressed zero-divisor graph, denoted by Γc(R), is the compression of a zero-divisor graph, in which the vertex set is the set of all equivalence classes of nonzero zero-divisors of a ring R, that is, the vertex set of Γc(R) is Re − {[0], [1]}, where Re = {[r] : r ∈ R} and two distinct equivalence classes [r] and [s] are adjacent if and only if rs = 0. In this article, we investigate the planarity of Γc(R) for some finite local rings of order p 2 , p 3 and determine the planarity of compressed zero-divisor graph of some local rings of order 32, whose zero-divisor graph is nonplanar. Further, we determine values of m and n for which Γc(Zn) and Γc(Zn[x]/(xm)) are planar.

Let R be a commutative ring with identity and A(R) be the set of ideals with nonzero annihilator. The annihilating-ideal graph of R is defined as the graph AG(R) with the vertex set A(R)* = A(R)\{0} and two distinct vertices I and J are adjacent if and only if IJ = 0. In this paper, we study the domination number of AG(R) and some connections between the domination numbers of annihilating-ideal graphs and zero-divisor graphs are given.

On zero-divisor graphs of small finite commutative rings

This article presents a survey of results consisting of the Wiener index of graphs associated with commutative rings. In particular, we focus on zero-divisor graphs, unit graphs, total graphs and prime graphs. Further, we have posed some open problems corresponding to the graphs considered.