On the regular graph of ideals of total ring of fractions

2018 ◽  
Vol 17 (07) ◽  
pp. 1850124
Author(s):  
Masoud Karimi
Keyword(s):  
Commutative Ring ◽  
Regular Graph ◽  
Zero Divisor ◽  
Independence Number ◽  
Graph Theoretic ◽  
Ring Of Fractions ◽  
Reduced Ring ◽  
Vertex Set ◽  
Basic Graph

The regular graph of ideals of a commutative ring [Formula: see text], denoted by [Formula: see text], is a graph whose vertex-set is the set of all nontrivial ideals of [Formula: see text] and, for every two distinct vertices [Formula: see text] and [Formula: see text], there is an edge between [Formula: see text] and [Formula: see text], whenever [Formula: see text] contains a nonzero zero-divisor on [Formula: see text] or vice versa. In this paper, we will show that [Formula: see text] is isomorphic to an induced sub-graph of [Formula: see text] which nicely can describe [Formula: see text]. This approach enables us to find the independence number of [Formula: see text] when [Formula: see text] is a reduced ring. Among other things, some basic graph theoretic properties of [Formula: see text] and relevant ring theoretic properties of [Formula: see text] will be studied.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

On the intersection graph of gamma sets in the zero-divisor graph

2015 ◽  
Vol 07 (01) ◽  
pp. 1450067 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Selvakumar
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Intersection Graph ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring. The intersection graph of gamma sets in the zero-divisor graph Γ(R) of R is the graph IΓ(R) with vertex set as the collection of all gamma sets of the zero-divisor graph Γ(R) of R and two distinct vertices A and B are adjacent if and only if A ∩ B ≠ ∅. In this paper, we study about various properties of IΓ(R) and investigate the interplay between the graph theoretic properties of IΓ(R) and the ring theoretic properties of R.

Some Results on Annihilating-ideal Graphs

2016 ◽  
Vol 59 (3) ◽  
pp. 641-651
Author(s):  
Farzad Shaveisi
Keyword(s):  
Direct Summand ◽  
Commutative Ring ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Complete Bipartite Graph ◽  
Independence Number ◽  
Prime Ideals ◽  
Reduced Ring ◽  
Vertex Set ◽  
Minimal Prime

AbstractThe annihilating-ideal graph of a commutative ring R, denoted by 𝔸𝔾(R), is a graph whose vertex set consists of all non-zero annihilating ideals and two distinct vertices I and J are adjacent if and only if IJ = (0). Here we show that if R is a reduced ring and the independence number of 𝔸𝔾(R) is finite, then the edge chromatic number of 𝔸𝔾(R) equals its maximum degree and this number equals 2|Min(R)|−1 also, it is proved that the independence number of 𝔸𝔾(R) equals 2|Min(R)|−1, where Min(R) denotes the set of minimal prime ideals of R. Then we give some criteria for a graph to be isomorphic with an annihilating-ideal graph of a ring. For example, it is shown that every bipartite annihilating-ideal graph is a complete bipartite graph with at most two horns. Among other results, it is shown that a ûnite graph 𝔸𝔾(R) is not Eulerian, and that it is Hamiltonian if and only if R contains no Gorenstain ring as its direct summand.

Some results on a spanning subgraph of the complement of the annihilating-ideal graph of a commutative reduced ring

2019 ◽  
Vol 11 (01) ◽  
pp. 1950012
Author(s):  
S. Visweswaran ◽  
Anirudhdha Parmar
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Undirected Graph ◽  
Ideal Formula ◽  
Graph Theoretic ◽  
Spanning Subgraph ◽  
Reduced Ring ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity which is not an integral domain. Let [Formula: see text] denote the set of all annihilating ideals of [Formula: see text] and let us denote [Formula: see text] by [Formula: see text]. For an ideal [Formula: see text] of [Formula: see text], we denote the annihilator of [Formula: see text] in [Formula: see text] by [Formula: see text]. That is, [Formula: see text]. In this note, for any ring [Formula: see text] with [Formula: see text], we associate an undirected graph denoted by [Formula: see text] whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] are joined by an edge if and only if either [Formula: see text] or [Formula: see text]. Let [Formula: see text] be a reduced ring. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

Annihilating-ideal graphs of commutative rings

2019 ◽  
Vol 13 (07) ◽  
pp. 2050121
Author(s):  
M. Aijaz ◽  
S. Pirzada
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Metric Dimension ◽  
Maximal Ideals ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with unity [Formula: see text]. The annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is defined to be the graph with vertex set [Formula: see text] — the set of non-zero annihilating ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] adjacent if and only if [Formula: see text]. Some connections between annihilating-ideal graphs and zero divisor graphs are given. We characterize the prime ideals (or equivalently maximal ideals) of [Formula: see text] in terms of their degrees as vertices of [Formula: see text]. We also obtain the metric dimension of annihilating-ideal graphs of commutative rings.

RECOVERING RINGS FROM ZERO-DIVISOR GRAPHS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350047 ◽  
Author(s):  
SHANE P. REDMOND
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Zero Divisor ◽  
Zero Divisor Graph ◽  
Reduced Ring ◽  
Non Local

Suppose G is the zero-divisor graph of some commutative ring with 1. When G has four or more vertices, a method is presented to find a specific commutative ring R with 1 such that Γ(R) ≅ G. Furthermore, this ring R can be written as R ≅ R1 × R2 × ⋯ × Rn, where each Ri is local and this representation of R is unique up to factors Ri with isomorphic zero-divisor graphs. It is also shown that for graphs on four or more vertices, no local ring has the same zero-divisor graph as a non-local ring and no reduced ring has the same zero-divisor graph as a non-reduced ring.

scholarly journals Thek-Zero-Divisor Hypergraph of a Commutative Ring

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Ch. Eslahchi ◽  
A. M. Rahimi
Keyword(s):  
Commutative Ring ◽  
Nilpotent Element ◽  
Zero Divisor ◽  
Clique Number ◽  
Common Point ◽  
Prime Ideals ◽  
Uniform Hypergraph ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

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