A Kind of Graph Structure on Non-reduced Rings

2010 ◽  
Vol 17 (01) ◽  
pp. 173-180 ◽  
Author(s):  
Aihua Li ◽  
Qisheng Li
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Maximum Degree ◽  
Integral Domain ◽  
Commutative Rings ◽  
Graph Structure ◽  
Graph Theoretic ◽  
Nilpotent Ring ◽  
Even Order ◽  
Finite Commutative Ring

In this paper, a kind of graph structure ΓN(R) of a ring R is introduced, and the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ΓN(R) is investigated. It is shown that if R is Artinian or commutative, then ΓN(R) is connected, the diameter of ΓN(R) is at most 3; and if ΓN(R) contains a cycle, then the girth of ΓN(R) is not more than 4; moreover, if R is non-reduced, then the girth of ΓN(R) is 3. For a finite commutative ring R, it is proved that the edge chromatic number of ΓN(R) is equal to the maximum degree of ΓN(R) unless R is a nilpotent ring with even order. It is also shown that, with two exceptions, if R is a finite reduced commutative ring and S is a commutative ring which is not an integral domain and ΓN(R) ≃ ΓN(S), then R ≃ S. If R and S are finite non-reduced commutative rings and ΓN(R) ≃ ΓN(S), then |R|=|S| and |N(R)|=|N(S)|.

A generalization of co-maximal graph of commutative rings

2019 ◽  
Vol 11 (01) ◽  
pp. 1950013
Author(s):  
B. Biswas ◽  
S. Kar ◽  
M. K. Sen ◽  
T. K. Dutta
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Commutative Rings ◽  
Regular Elements ◽  
Graph Theoretic ◽  
Finite Commutative Ring

Let [Formula: see text] be a commutative ring with [Formula: see text]. Let [Formula: see text] be a graph with vertices as elements of [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] for some non-zero idempotent [Formula: see text] in [Formula: see text]. In this paper, we establish a relation between completeness of the graph [Formula: see text] and regularity of the ring [Formula: see text]. For a finite commutative ring [Formula: see text] with [Formula: see text], we show that the chromatic number of [Formula: see text] is equal to the number of regular elements in [Formula: see text]. Moreover, we characterize some graph theoretic properties of [Formula: see text] and finally we characterize Eulerian property of the graph [Formula: see text].

scholarly journals Line graph of unit graphs associated with finite commutative rings

2021 ◽  
Author(s):  
Pranjali ◽  
Amit Kumar ◽  
Pooja Sharma
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Line Graph ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Common Vertex ◽  
Unit Graph ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Nonzero Identity

For a given graph G, its line graph denoted by L(G) is a graph whose vertex set V (L(G)) = E(G) and {e1, e2} ∈ E(L(G)) if e1 and e2 are incident to a common vertex in G. Let R be a finite commutative ring with nonzero identity and G(R) denotes the unit graph associated with R. In this manuscript, we have studied the line graph L(G(R)) of unit graph G(R)  associated with R. In the course of the investigation, several basic properties, viz., diameter, girth, clique, and chromatic number of L(G(R)) have been determined. Further, we have derived sufficient conditions for L(G(R)) to be Planar and Hamiltonian

scholarly journals On irreducible divisor graphs in commutative rings with zero-divisors

2015 ◽  
Vol 46 (4) ◽  
pp. 365-388
Author(s):  
Christopher Park Mooney
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Zero Divisors ◽  
Classical Paper

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in integral domains. This inspired the so called irreducible divisor graph of an integral domain studied by J. Coykendall and J. Maney. Factorization in rings with zero-divisors is considerably more complicated than integral domains and has been widely studied recently. We find that many of the same techniques can be extended to rings with zero-divisors. In this article, we construct several distinct irreducible divisor graphs of a commutative ring with zero-divisors. This allows us to use graph theoretic properties to help characterize finite factorization properties of commutative rings, and conversely.

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).

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.

The Classification of the Annihilating-Ideal Graphs of Commutative Rings

2014 ◽  
Vol 21 (02) ◽  
pp. 249-256 ◽  
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 ◽  
Vertex Set

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

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.

Orthogonal decompositions of Lie algebras over finite commutative rings

2021 ◽  
pp. 2350006
Author(s):  
Songpon Sriwongsa
Keyword(s):  
Commutative Ring ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Commutative Rings ◽  
Orthogonal Decomposition ◽  
Necessary Condition ◽  
Special Linear ◽  
Orthogonal Lie Algebra ◽  
Orthogonal Decompositions ◽  
Finite Commutative Ring

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

On zero-divisor graphs of commutative rings without identity

2019 ◽  
Vol 19 (12) ◽  
pp. 2050226 ◽  
Author(s):  
G. Kalaimurugan ◽  
P. Vignesh ◽  
T. Tamizh Chelvam
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Free Order ◽  
Finite Commutative Rings

Let [Formula: see text] be a finite commutative ring without identity. In this paper, we characterize all finite commutative rings without identity, whose zero-divisor graphs are unicyclic, claw-free and tree. Also, we obtain all finite commutative rings without identity and of cube-free order for which the corresponding zero-divisor graph is toroidal.

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