The essential ideal graph of a commutative ring

Let [Formula: see text] be a commutative ring with identity. The essential ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] is an essential ideal. In this paper, we initiate the study of the essential ideal graph of a commutative ring and we investigate its properties.

Finite commutative ring with genus two essential graph

10.1142/s0219498818501219 ◽

2018 ◽

Vol 17 (07) ◽

pp. 1850121

Author(s):

K. Selvakumar ◽

M. Subajini ◽

M. J. Nikmehr

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Zero Divisors ◽

Essential Ideal ◽

Vertex Set ◽

Finite Commutative Ring ◽

Genus Two ◽

Finite Commutative Rings

Let [Formula: see text] be a commutative ring with identity and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity for which the genus of [Formula: see text] is two.

Crosscap two of class of graphs from commutative rings

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830922500410 ◽

2021 ◽

Author(s):

S. Karthik ◽

S. N. Meera ◽

K. Selvakumar

Keyword(s):

Commutative Ring ◽

Undirected Graph ◽

Commutative Rings ◽

Zero Divisors ◽

Essential Ideal ◽

Vertex Set ◽

Annihilator Graph ◽

Finite Commutative Rings

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The annihilator graph of commutative ring [Formula: see text] is the simple undirected graph [Formula: see text] with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is an essential ideal. In this paper, we classify all finite commutative rings with identity whose annihilator graph and essential graph have crosscap two.

On the essential graph of a commutative ring

10.1142/s0219498817501328 ◽

2016 ◽

Vol 16 (07) ◽

pp. 1750132 ◽

Cited By ~ 1

Author(s):

M. J. Nikmehr ◽

R. Nikandish ◽

M. Bakhtyiari

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Zero Divisor ◽

Artinian Ring ◽

Clique Number ◽

Zero Divisor Graph ◽

Zero Divisors ◽

Essential Ideal ◽

Let [Formula: see text] be a commutative ring with identity, and let [Formula: see text] be the set of zero-divisors of [Formula: see text]. The essential graph of [Formula: see text] is defined as the graph [Formula: see text] with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if ann[Formula: see text] is an essential ideal. It is proved that [Formula: see text] is connected with diameter at most three and with girth at most four, if [Formula: see text] contains a cycle. Furthermore, rings with complete or star essential graphs are characterized. Also, we study the affinity between essential graph and zero-divisor graph that is associated with a ring. Finally, we show that the essential graph associated with an Artinian ring is weakly perfect, i.e. its vertex chromatic number equals its clique number.

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

10.1142/s021949881250199x ◽

2013 ◽

Vol 12 (04) ◽

pp. 1250199 ◽

Cited By ~ 11

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).

The Weakly Nilpotent Graph of a Commutative Ring

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-096-1 ◽

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 ◽

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

10.1142/s0219498812501794 ◽

2012 ◽

Vol 12 (03) ◽

pp. 1250179 ◽

Cited By ~ 10

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.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

Algebra Colloquium ◽

10.1142/s1005386714000200 ◽

2014 ◽

Vol 21 (02) ◽

pp. 249-256 ◽

Cited By ~ 20

Author(s):

G. Aalipour ◽

S. Akbari ◽

M. Behboodi ◽

R. Nikandish ◽

M. J. Nikmehr ◽

...

Keyword(s):

Commutative Ring ◽

Chromatic Number ◽

Artinian Ring ◽

Clique Number ◽

Commutative Rings ◽

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. 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). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

A generalized ideal-based zero-divisor graph

10.1142/s0219498815500796 ◽

2015 ◽

Vol 14 (06) ◽

pp. 1550079 ◽

Cited By ~ 1

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 ◽

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

Asian-European Journal of Mathematics ◽

10.1142/s1793557120501211 ◽

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 ◽

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.

The proof of a conjecture in Jacobson graph of a commutative ring

10.1142/s0219498815501078 ◽

2015 ◽

Vol 14 (10) ◽

pp. 1550107 ◽

Cited By ~ 2

Author(s):

S. Akbari ◽

S. Khojasteh ◽

A. Yousefzadehfard

Keyword(s):

Commutative Ring ◽

Local Ring ◽

Clique Number ◽

Finite Ring ◽

Jacobson Graph ◽

Vertex Set ◽

Nonzero Identity

Let R be a commutative ring with nonzero identity. The Jacobson graph of R denoted by 𝔍R is a graph with the vertex set R\J(R), and two distinct vertices x, y ∈ V(𝔍R) are adjacent if and only if 1 - xy ∉ U(R), where U(R) is the set of all unit elements of R. Let ω(𝔍R) denote the clique number of 𝔍R. It was conjectured that if [Formula: see text] is a commutative finite ring and (Ri, 𝔪i) is a local ring, for i = 1, …, n, then [Formula: see text], where Fi = Ri/𝔪i, for i = 1, …, n. In this paper, we prove that if R is a commutative ring (not necessarily finite) and R is not a field, then ω(𝔍R) = max 𝔪∈ Max (R) |𝔪| and using this we show that the aforementioned conjecture holds.