On the planar, outer planar, cut vertices and end-regular comaximal graph of lattices

2015 ◽  
Vol 86 (3-4) ◽  
pp. 295-312
Author(s):  
MOJGAN AFKHAMI ◽  
KAZEM KHASHYARMANESH ◽  
FAEZE SHAHSAVAR
Keyword(s):  
Comaximal Graph

scholarly journals When the Comaximal Graph of a Lattice is Toroidal

2016 ◽  
Vol 56 (3) ◽  
pp. 683-714
Author(s):  
Mojgan Afkhami ◽  
Khadijeh Ahmad Javaheri ◽  
Kazem Khashyarmanesh
Keyword(s):  
Comaximal Graph

When is the Comaximal Graph Split?

2012 ◽  
Vol 40 (7) ◽  
pp. 2400-2404 ◽  
Author(s):  
M. I. Jinnah ◽  
Shine C. Mathew
Keyword(s):  
Comaximal Graph

The annihilators comaximal graph

2021 ◽  
Author(s):  
Saeed Rajaee
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Prime Numbers ◽  
Graph Theoretic ◽  
Special Cases ◽  
Unitary Module ◽  
Comaximal Graph

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

scholarly journals A Note on Comaximal Graph of Non-commutative Rings

2011 ◽  
Vol 16 (2) ◽  
pp. 303-307 ◽  
Author(s):  
Saieed Akbari ◽  
Mohammad Habibi ◽  
Ali Majidinya ◽  
Raoofe Manaviyat
Keyword(s):  
Commutative Rings ◽  
Comaximal Graph

On the Comaximal Graph of a Commutative Ring

2014 ◽  
Vol 57 (2) ◽  
pp. 413-423 ◽  
Author(s):  
Karim Samei
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Topological Properties ◽  
Dominating Sets ◽  
Graph Properties ◽  
Comaximal Graph

AbstractLet R be a commutative ring with 1. In a 1995 paper in J. Algebra, Sharma and Bhatwadekar defined a graph on R, Γ(R), with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. In this paper, we consider a subgraph Γ2(R) of Γ(R) that consists of non-unit elements. We investigate the behavior of Γ2(R) and Γ2(R)\J(R), where J(R) is the Jacobson radical of R. We associate the ring properties of R, the graph properties of Γ(R), and the topological properties of Max(R). Diameter, girth, cycles and dominating sets are investigated, and algebraic and topological characterizations are given for graphical properties of these graphs.

scholarly journals Independence complexes of comaximal graphs of commutative rings with identity

2015 ◽  
Vol 98 (112) ◽  
pp. 109-117
Author(s):  
Nela Milosevic
Keyword(s):  
Commutative Rings ◽  
Homotopy Equivalent ◽  
Comaximal Graph ◽  
Independence Complex

We study topology of the independence complexes of comaximal (hyper)graphs of commutative rings with identity. We show that the independence complex of comaximal hypergraph is contractible or homotopy equivalent to a sphere, and that the independence complex of comaximal graph is almost always contractible.

scholarly journals Comaximal graph of $C(X)$

2016 ◽  
Vol 57 (3) ◽  
pp. 353-364
Author(s):  
 Badie Mehdi
Keyword(s):  
Comaximal Graph

On the projective comaximal graphs of lattices

2016 ◽  
Vol 08 (04) ◽  
pp. 1650072
Author(s):  
A. Parsapour ◽  
KH. Ahmad Javaheri
Keyword(s):  
Finite Lattice ◽  
Comaximal Graph

In this paper, we investigate the projectivity of the comaximal graph of a finite lattice.

scholarly journals When Is the Complement of the Comaximal Graph of a Commutative Ring Planar?

ISRN Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
S. Visweswaran ◽  
Jaydeep Parejiya
Keyword(s):  
Commutative Ring ◽  
Jacobson Radical ◽  
Comaximal Graph

Let R be a commutative ring with identity. In this paper we classify rings R such that the complement of comaximal graph of R is planar. We also consider the subgraph of the complement of comaximal graph of R induced on the set S of all nonunits of R with the property that each element of S is not in the Jacobson radical of R and classify rings R such that this subgraph is planar.

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