Divisor Graph Research Articles

The Zero Divisor Graphs of Commutative Local Rings of Order p 4 and p 5

To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4] these classes include all local rings of order p 4 and p 5 for prime p.

A class of finite groups with complete character degree graphs

In this paper, we construct a new class of finite groups whose common divisor graphs are complete graphs, while there is no prime dividing all the nontrivial degrees.

On generalized zero divisor graph of a poset

In this paper, we introduce the generalized ideal based zero divisor graph of a poset P, denoted by GI(P)̂. A representation theorem is obtained for generalized zero divisor graphs. It is proved that a graph is complete r-partite with r⩾2 if and only if it is a generalized zero divisor graph of a poset. As a consequence of this result, we prove a form of a Beck’s Conjecture for generalized zero divisor graphs of a poset. Further, it is proved that a generalized zero divisor graph G{0}(P)̂ of a section semi-complemented poset P with respect to the ideal (0] is a complete graph.

The binding number of a zero divisor graph

In this paper, we evaluate b(Γ(Zn)). Our main result is, we give maximum value of b(Γ(Zn)) is 0.99999999796427626489236243072661, where n is any positive integer upto fiftieth million. Mathematics Subject Classification: 05C25, 05C35, 05C70

Rectilinear crossing number of a zero divisor graph

In this paper, we evaluate the rectilinear crossing number of Γ(Zn). We mainly focus on finding the rectilinear drawing of zero divisor complete graphs especially for p =7 , 11. Finally, we compare the rectilinear crossing number with the crossing number thereby forming an inequality by different conjectures.

Irreducible divisor graphs for numerical monoids

The factorization of an element x from a numerical monoid can be represented visually as an irreducible divisor graph G.x/. The vertices of G.x/ are the monoid generators that appear in some representation of x, with two vertices adjacent if they both appear in the same representation. In this paper, we determine precisely when irreducible divisor graphs of elements in monoids of the form NDhn; nC 1;:::; nC ti where 0 t < n are complete, connected, or have a maximum number of vertices. Finally, we give examples of irreducible divisor graphs that are isomorphic to each of the 31 mutually nonisomorphic connected graphs on at most five vertices.

ZERO DIVISOR GRAPHS OF LATTICES AND PRIMAL IDEALS

In this paper, we introduce the concepts of primal and weakly primal ideals in lattices. Further, the diameter of the zero divisor graph of a lattice with respect to a non-primal ideal is determined.

BIPARTITE DIVISOR GRAPH FOR THE PRODUCT OF SUBSETS OF INTEGERS

AbstractThe bipartite divisor graph B(X), for a set Xof positive integers, and some of its properties have recently been studied. We construct the bipartite divisor graph for the product of subsets of positive integers and investigate some of its properties. We also give some applications in group theory.

Planar zero divisor graphs of partially ordered sets

We associate a graph G∗(P) to a partially ordered set (poset, briefly) with the least element 0, as an undirected graph with vertex set P∗=P∖{0} and, for two distinct vertices x and y, x is adjacent to y in G∗(P) if and only if {x,y}l={0}, where, for a subset S of P, Sl is the set of all elements x∈P with x≦s for all s∈S. We study some basic properties of G∗(P). Also, we completely investigate the planarity of G∗(P).

The Complement Graph for Gaussian Integers Modulo n

This article is a continuation for the work done in [1, 2] on the zero divisor graph for the ring of Gaussian integers modulo n. It investigates when the complement graph of the zero divisor graph for the Gaussian integers modulo n connected, planar, regular, or Eulerian. The girth and diameter were also studied.

On the zero divisor graphs of [formula omitted]-lattices

In this paper, we study the graph theoretic properties of the zero divisor graph Γ(L) of a pm-lattice L. We have characterized the diameter and the eccentricity of Γ(L) when L is a semiprimitive pm-lattice. Further, an algebraic and a topological characterization is given for the graph Γ(L) to be triangulated or hyper-triangulated.

On the Line Graph of the Zero Divisor Graph for the Ring of Gaussian Integers Modulo n

Let Γ(ℤn[i]) be the zero divisor graph for the ring of the Gaussian integers modulo n. Several properties of the line graph of Γ(ℤn[i]), L(Γ(ℤn[i])) are studied. It is determined when L(Γ(ℤn[i])) is Eulerian, Hamiltonian, or planer. The girth, the diameter, the radius, and the chromatic and clique numbers of this graph are found. In addition, the domination number of L(Γ(ℤn[i])) is given when n is a power of a prime. On the other hand, several graph invariants for Γ(ℤn[i]) are also determined.

On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n

The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally and locally connected is given. The chromatic number when is a power of a prime is computed. Further properties for and are also discussed.

Irreducible Divisor Graphs and Factorization Properties of Domains

This article examines the connections between the factorization properties of a domain, e.g., unique factorization domain (UFD), finite factorization domain (FFD), and the domain's irreducible divisor graphs. In particular, we show that although there are some nice correlations between the properties of the domain D and the set of irreducible divisor graphs {G(x): x ∈ D* \\ U(D)} when D is an FFD, it is very unlikely that any information about the domain D can be gleaned from the collection {G(x): x ∈ D* \\ U(D)} when D is not an FFD. We also introduce an alternate irreducible divisor graph called the compressed irreducible divisor graph and study some of its properties.

A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors

We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph.

Zero Divisor Graph of a Poset with Respect to an Ideal

In this paper, we introduce the zero divisor graph GI(P) of a poset P (with 0) with respect to an ideal I. It is shown that GI(P) is connected with its diameter ≤3, and if GI(P) contains a cycle, then the core K of GI(P) is a union of 3-cycles and 4-cycles. Further, the chromatic number and clique number of GI(P) are shown to be equal. This proves a form of Beck’s conjecture for posets with 0. The complete bipartite zero divisor graphs are characterized.

Energy and Wiener Index of Zero-Divisor Graphs

Let R be a commutative ring and (R) be its zerodivisor graph. In this article, we study Wiener index and energy of Γ(Zn ) where n = pq or n = p 2 q and p, q are primes. A MATLAB

SOME PROPERTIES OF THE ZERO-DIVISOR GRAPH FOR THE RING OF GAUSSIAN INTEGERS MODULO n

AbstractThis paper is a continuation for the study of the zero-divisor graph for the ring of Gaussian integers modulo n, Γ(ℤn[i]) in [8] (Emad Abu Osba, Salah Al-Addasi and Nafez Abu Jaradeh. Zero divisor graph for the ring of Gaussin integers modulo n. Comm. Algebra 36(10) (2008), 3865–3877). It is investigated, when is Γ(ℤn[i]) locally H, Hamiltonian or bipartite graph? A full characterisation for the chromatic number and the radius is also given.

Analysis of Adjacency Matrix and Neighborhood Associated with Zero Divisor Graph of Finite Commutative Rings

The present investigation is concerned with zero divisor graph of direct Product of finite commutative rings and to give some new ideas about its corresponding adjacency matrix. In the first section of the paper, we study about neighborhood set of the zero divisor graph of direct product over finite commutative rings. In the second section we discussed some examples of these ring. Finally, some surprsing results (regarding to the adjacency matrix) and theorems also estabilised. The study on neighborhood may be used to represent graph in computer algorthim, neighborhood are used to determine the clustering cofficient of graph and adjacency martix is useful in computer application.

Local Rings with Genus Two Zero Divisor Graph

To each commutative ring R we can associate a graph, the zero divisor graph of R, whose vertices are the zero divisors of R, and such that two vertices are adjacent if their product is zero. Presently, we enumerate the local finite commutative rings whose zero divisor graphs have orientable genus 2.