scispace - formally typeset
Search or ask a question

Some properties and domination number of the complement of a new graph associated to a commutative ring

01 Jan 2018-Vol. 3, Iss: 12, pp 99-108
TL;DR: It is shown that ΩR∗ is a connected graph with diam(ΩR ∼3) ≤ 3 unless R is isomorphic to a direct product of two fields and all commutative rings R with at least two maximal ideals for which Ω R∗ are planar are characterized.
Abstract: In this paper some properties of the complement of a new graph associated with a commutative ring are investigated ....
References
More filters
Journal ArticleDOI
TL;DR: This paper initiates the study of the co-annihilating graph of a commutative ring and investigates its properties.
Abstract: Let R be a commutative ring with identity and 𝔘R be the set of all non-zero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜R, is a graph with vertex set 𝔘R and two vertices a and b are adjacent whenever Ann(a) ∩Ann(b) = (0). In this paper, we initiate the study of the co-annihilating graph of a commutative ring and we investigate its properties.

2 citations

DOI
01 Sep 2017
TL;DR: Alilou, Amjadi and Sheikholeslami as discussed by the authors investigated a graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either ann(I)J = (0) or ann(J)I = (1).
Abstract: The rings considered in this article are commutative with identity which are not fields. Let R be a ring. A. Alilou, J. Amjadi and Sheikholeslami introduced and investigated a graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either ann(I)J = (0) or ann(J)I = (0). They called this graph as a new graph associated to a commutative ring.Their above mentioned work appeared in the Journal, Discrete Mathematics Algorithms and Applications. The aim of this article is to investigate the interplay between some graph- theoretic properties of the complement of a new graph associated to a commutative ring R and the ring -theoretic-properties of R.

2 citations


"Some properties and domination numb..." refers background or methods in this paper

  • ...The annihilatingideal graph of a commutative ring and other related graphs have been studied by several researchers (see for example, [1, 2, 14, 18, 19])....

    [...]

  • ...Several interesting and inspiring theorems were proved on ΩR in [2] (see for example, Theorems 4, 10, and 20)....

    [...]

  • ...Motivated by the work done on the annihilating-ideal graph of a commutative ring, in [2], Alilou, Amjadi and Sheikholeslami introduced and studied a new graph associated to a commutative ring R, denoted by ΩR, which is an undirected graph whose vertex set is the set of all nontrivial ideals of R and distinct vertices I, J are joined by an edge in this graph if and only if either (AnnRI)J = (0) or (AnnRJ)I = (0) (that is, if and only if either AnnRI ⊆ AnnRJ or AnnRJ ⊆ AnnRI), where for an ideal I of R, the annihilator of I in R, denoted by AnnRI is defined as AnnRI = {r ∈ R : Ir = (0)}....

    [...]

Journal ArticleDOI
TL;DR: The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-the theoretical properties of [/Formula]: see text.
Abstract: Let R be a commutative ring with identity which is not an integral domain. Let 𝔸(R) denote the set of all annihilating ideals of R and let us denote 𝔸(R)\{(0)} by 𝔸(R)∗. For an ideal I of R, we den...

1 citations

Journal ArticleDOI
TL;DR: This paper characterize all Artinian rings [Formula: see text] for which the genus of [Form formula]: see text is zero or one.
Abstract: 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 nonzero proper ideals of R and two distinct vertices I ...

1 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the planar and outerplanar indices of some graphs associated to a commutative ring and gave a full characterization of these graphs with respect to their planar indices when R is a finite ring.
Abstract: Abstract In this paper, we study the planar and outerplanar indices of some graphs associated to a commutative ring. We give a full characterization of these graphs with respect to their planar and outerplanar indices when R is a finite ring.

1 citations