On the Annihilator Graph of a Commutative Ring

2013 ◽  
Vol 42 (1) ◽  
pp. 108-121 ◽  
Author(s):  
Ayman Badawi
Keyword(s):  
Commutative Ring ◽  
Annihilator Graph

Some Properties of Annihilator Graph of a Commutative Ring

2014 ◽  
Vol 10 (5) ◽  
pp. 61-68
Author(s):  
Priyanka Pratim Baruah ◽  
◽  
Kuntala Patra
Keyword(s):  
Commutative Ring ◽  
Annihilator Graph

Crosscap two of class of graphs from commutative rings

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.

Some Remarks on the Annihilator Graph of a Commutative Ring

2020 ◽  
Vol 49 (2) ◽  
pp. 325-332
Author(s):  
M. ADLIFARD ◽  
Sh. PAYROVI
Keyword(s):  
Commutative Ring ◽  
Annihilator Graph

More on the annihilator graph of a commutative ring

2017 ◽  
Vol 46 (1) ◽  
pp. 107-118 ◽  
Author(s):  
M. J. NIKMEHR ◽  
R. NIKANDISH ◽  
M. BAKHTYIARI
Keyword(s):  
Commutative Ring ◽  
Annihilator Graph

The metric dimension of annihilator graphs of commutative rings

2019 ◽  
Vol 19 (05) ◽  
pp. 2050089
Author(s):  
V. Soleymanivarniab ◽  
A. Tehranian ◽  
R. Nikandish
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Zero Divisors ◽  
Vertex Set ◽  
Annihilator Graph ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. The annihilator graph of [Formula: see text], denoted by [Formula: see text], is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of [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 study the metric dimension of annihilator graphs associated with commutative rings and some metric dimension formulae for annihilator graphs are given.

Coloring of the annihilator graph of a commutative ring

2016 ◽  
Vol 15 (07) ◽  
pp. 1650124 ◽  
Author(s):  
R. Nikandish ◽  
M. J. Nikmehr ◽  
M. Bakhtyiari
Keyword(s):  
Commutative Ring ◽  
Explicit Formula ◽  
Chromatic Number ◽  
Clique Number ◽  
Zero Divisors ◽  
Vertex Set ◽  
Annihilator Graph

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 annihilator graph of [Formula: see text] is defined as the graph AG[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]. In this paper, we study annihilator graphs of rings with equal clique number and chromatic number. For some classes of rings, we give an explicit formula for the clique number of annihilator graphs. Among other results, bipartite annihilator graphs of rings are characterized. Furthermore, some results on annihilator graphs with finite clique number are given.

scholarly journals When the annihilator graph of a commutative ring is planar or toroidal?

2020 ◽  
Vol 24 (2) ◽  
pp. 281-290
Author(s):  
Moharram Bakhtyiari ◽  
Reza Nikandish ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Annihilator Graph

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The annihilator graph of R is defined as the undirected graph AG(R) with the vertex set Z(R)* = Z(R) \ {0}, and two distinct vertices x and y are adjacent if and only if  ann_R(xy) \neq ann_R(x) \cup ann_R(y). In this paper, all rings whose annihilator graphs can be embedded on the plane or torus are classified.

On perfect annihilator graphs of commutative rings

2020 ◽  
Vol 12 (05) ◽  
pp. 2050060
Author(s):  
Sh. Ebrahimi ◽  
A. Tehranian ◽  
R. Nikandish
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Zero Divisors ◽  
Vertex Set ◽  
Annihilator Graph ◽  
Vast Range

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 annihilator 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]. In this paper, we study the perfectness of annihilator graphs of a vast range of rings. Indeed, it is shown that if [Formula: see text] is reduced with finitely many minimal primes or nonreduced, then [Formula: see text] is perfect.

Some results on the annihilator graph of~a~commutative ring

2017 ◽  
Vol 67 (1) ◽  
pp. 151-169 ◽  
Author(s):  
Mojgan Afkhami ◽  
Kazem Khashyarmanesh ◽  
Zohreh Rajabi
Keyword(s):  
Commutative Ring ◽  
Annihilator Graph
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