On the Annihilator Graph of a Commutative Ring

2013 ◽  
Vol 42 (1) ◽  
pp. 108-121 ◽  
Author(s):  
Ayman Badawi
2014 ◽  
Vol 10 (5) ◽  
pp. 61-68
Author(s):  
Priyanka Pratim Baruah ◽  
◽  
Kuntala Patra

Author(s):  
S. Karthik ◽  
S. N. Meera ◽  
K. Selvakumar

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.


2020 ◽  
Vol 49 (2) ◽  
pp. 325-332
Author(s):  
M. ADLIFARD ◽  
Sh. PAYROVI

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

2019 ◽  
Vol 19 (05) ◽  
pp. 2050089
Author(s):  
V. Soleymanivarniab ◽  
A. Tehranian ◽  
R. Nikandish

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.


2016 ◽  
Vol 15 (07) ◽  
pp. 1650124 ◽  
Author(s):  
R. Nikandish ◽  
M. J. Nikmehr ◽  
M. Bakhtyiari

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.


2020 ◽  
Vol 24 (2) ◽  
pp. 281-290
Author(s):  
Moharram Bakhtyiari ◽  
Reza Nikandish ◽  
Mohammad Javad Nikmehr

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.


2020 ◽  
Vol 12 (05) ◽  
pp. 2050060
Author(s):  
Sh. Ebrahimi ◽  
A. Tehranian ◽  
R. Nikandish

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.


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

Sign in / Sign up

Export Citation Format

Share Document