scholarly journals On the Nonorientable Genus of the Generalized Unit and Unitary Cayley Graphs of a Commutative Ring

2022 ◽  
Vol 29 (01) ◽  
pp. 167-180
Author(s):  
Mahdi Reza Khorsandi ◽  
Seyed Reza Musawi
Keyword(s):  
Commutative Ring ◽  
Multiplicative Group ◽  
Cayley Graphs ◽  
Artinian Ring ◽  
Finite Rings ◽  
Artinian Rings ◽  
Vertex Set ◽  
Genus Formula ◽  
Multiplicative Subgroup ◽  
Unitary Cayley Graph

Let [Formula: see text] be a commutative ring and [Formula: see text] the multiplicative group of unit elements of [Formula: see text]. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, [Formula: see text], corresponding to a multiplicative subgroup [Formula: see text] of [Formula: see text] and a nonempty subset [Formula: see text] of [Formula: see text] with [Formula: see text], as the graph with vertex set [Formula: see text]and two distinct vertices [Formula: see text] and [Formula: see text] being adjacent if and only if there exists [Formula: see text] such that [Formula: see text]. In this paper, we characterize all Artinian rings [Formula: see text] for which [Formula: see text] is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring [Formula: see text] for which [Formula: see text] has finite nonorientable genus, [Formula: see text] must be a finite ring. Finally, it is proved that for a given positive integer [Formula: see text], the number of finite rings [Formula: see text] for which [Formula: see text] has nonorientable genus [Formula: see text] is finite.

scholarly journals RINGS WHOSE ANNIHILATING-IDEAL GRAPHS HAVE POSITIVE GENUS

2012 ◽  
Vol 11 (03) ◽  
pp. 1250049 ◽  
Author(s):  
F. ALINIAEIFARD ◽  
M. BEHBOODI
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Artinian Ring ◽  
Commutative Rings ◽  
Gorenstein Ring ◽  
Isomorphism Classes ◽  
Artinian Rings ◽  
Vertex Set ◽  
Positive Genus

Let R be a commutative ring and 𝔸(R) be the set of ideals with nonzero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). We investigate commutative rings R whose annihilating-ideal graphs have positive genus γ(𝔸𝔾(R)). It is shown that if R is an Artinian ring such that γ(𝔸𝔾(R)) < ∞, then either R has only finitely many ideals or (R, 𝔪) is a Gorenstein ring with maximal ideal 𝔪 and v.dimR/𝔪𝔪/𝔪2= 2. Also, for any two integers g ≥ 0 and q > 0, there are only finitely many isomorphism classes of Artinian rings R satisfying the conditions: (i) γ(𝔸𝔾(R)) = g and (ii) |R/𝔪| ≤ q for every maximal ideal 𝔪 of R. Also, it is shown that if R is a non-domain Noetherian local ring such that γ(𝔸𝔾(R)) < ∞, then either R is a Gorenstein ring or R is an Artinian ring with only finitely many ideals.

scholarly journals On Perfect Co-Annihilating-Ideal Graph of a Commutative Artinian Ring

2021 ◽  
Vol 45 (01) ◽  
pp. 63-73
Author(s):  
S. M. SAADAT MIRGHADIM ◽  
M. J. NIKMEHR ◽  
R. NIKANDISH
Keyword(s):  
Commutative Ring ◽  
Artinian Ring ◽  
Artinian Rings ◽  
Vertex Set

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by AR, is a graph whose vertex set is the set of all non-zero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) ∩ Ann(J) = (0). In this paper, we characterize all Artinian rings for which both of the graphs AR and AR (the complement of AR), are chordal. Moreover, all Artinian rings whose AR (and thus AR) is perfect are characterized.

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 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.

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

A characterization of rings whose unitary Cayley graphs are planar

2018 ◽  
Vol 17 (09) ◽  
pp. 1850178 ◽  
Author(s):  
Huadong Su ◽  
Yiqiang Zhou
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

Classification of rings with toroidal co-annihilating-ideal graphs

2020 ◽  
pp. 2150006
Author(s):  
Rana Khoeilar ◽  
Jafar Amjadi
Keyword(s):  
Commutative Ring ◽  
Artinian Rings ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity. The co-annihilating-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 distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text]. In this paper, we study the planarity and genus of [Formula: see text]. In particular, we characterize all Artinian rings [Formula: see text] for which the genus of [Formula: see text] is zero or one.

On the essential graph of a commutative ring

2016 ◽  
Vol 16 (07) ◽  
pp. 1750132 ◽  
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 ◽  
Vertex Set

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.

On the nil-graph of ideals of commutative Artinian rings

2018 ◽  
Vol 10 (05) ◽  
pp. 1850062
Author(s):  
T. Tamizh Chelvam ◽  
K. Selvakumar ◽  
P. Subbulakshmi
Keyword(s):  
Commutative Ring ◽  
Artinian Rings ◽  
Nilpotent Elements ◽  
Genus 2 ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring with identity and Nil[Formula: see text] be the ideal consisting of all nilpotent elements of [Formula: see text]. Let [Formula: see text] [Formula: see text] [Formula: see text]. The nil-graph of ideals of [Formula: see text] is defined as the graph [Formula: see text] whose vertex set is the set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we discuss some properties of nil-graph of ideals concerning connectedness, split and claw free. Also we characterize all commutative Artinian rings [Formula: see text] for which the nil-graph [Formula: see text] has genus 2.

scholarly journals Dot product graphs and domination number

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
Dina Saleh ◽  
Nefertiti Megahed
Keyword(s):  
Commutative Ring ◽  
Upper Bound ◽  
Undirected Graph ◽  
Multiplicative Group ◽  
Domination Number ◽  
Product Graph ◽  
Group Of Units ◽  
Product Graphs ◽  
Vertex Set ◽  
Dot Product

Abstract Let A be a commutative ring with 1≠0 and R=A×A. The unit dot product graph of R is defined to be the undirected graph UD(R) with the multiplicative group of units in R, denoted by U(R), as its vertex set. Two distinct vertices x and y are adjacent if and only if x·y=0∈A, where x·y denotes the normal dot product of x and y. In 2016, Abdulla studied this graph when $A=\mathbb {Z}_{n}$ A = ℤ n , $n \in \mathbb {N}$ n ∈ ℕ , n≥2. Inspired by this idea, we study this graph when A has a finite multiplicative group of units. We define the congruence unit dot product graph of R to be the undirected graph CUD(R) with the congruent classes of the relation $\thicksim $ ∽ defined on R as its vertices. Also, we study the domination number of the total dot product graph of the ring $R=\mathbb {Z}_{n}\times... \times \mathbb {Z}_{n}$ R = ℤ n × ... × ℤ n , k times and k<∞, where all elements of the ring are vertices and adjacency of two distinct vertices is the same as in UD(R). We find an upper bound of the domination number of this graph improving that found by Abdulla.

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