scholarly journals Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n

Algebra ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Osama Alkam ◽  
Emad Abu Osba
Keyword(s):  
Zero Divisor ◽  
Zero Divisor Graph

Let En be the ring of Eisenstein integers modulo n. In this paper we study the zero divisor graph Γ(En). We find the diameters and girths for such zero divisor graphs and characterize n for which the graph Γ(En) is complete, complete bipartite, bipartite, regular, Eulerian, Hamiltonian, or chordal.

scholarly journals Sub-semigroups determined by the zero-divisor graph

2008 ◽  
Vol 308 (22) ◽  
pp. 5122-5135 ◽  
Author(s):  
Tongsuo Wu ◽  
Dancheng Lu
Keyword(s):  
Zero Divisor ◽  
Zero Divisor Graph

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Satisfy Condition ◽  
Induced Subgraph ◽  
Zero Divisor Graph ◽  
Ring Graph ◽  
Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

A generalized ideal-based zero-divisor graph

2015 ◽  
Vol 14 (06) ◽  
pp. 1550079 ◽  
Author(s):  
M. J. Nikmehr ◽  
S. Khojasteh
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Chromatic Number ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Proper Ideal ◽  
Graph Structure ◽  
Zero Divisor Graph ◽  
Vertex Set

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

Zero-divisor Graph Generalizations

2021 ◽  
pp. 239-292
Author(s):  
David F. Anderson ◽  
T. Asir ◽  
Ayman Badawi ◽  
T. Tamizh Chelvam
Keyword(s):  
Zero Divisor ◽  
Zero Divisor Graph

On two extensions of the classical zero-divisor graph

2019 ◽  
Vol 14 (1) ◽  
pp. 341-350
Author(s):  
Malik Bataineh ◽  
Driss Bennis ◽  
Jilali Mikram ◽  
Fouad Taraza
Keyword(s):  
Zero Divisor ◽  
Zero Divisor Graph

Some properties of zero divisor graph obtained by the ring Zp × Zq × Zr

2019 ◽  
Vol 12 (06) ◽  
pp. 2040001
Author(s):  
Nihat Akgunes ◽  
Yasar Nacaroglu
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Topological Indices ◽  
Zero Divisor Graph

The concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra 116 (1988) 208–226]. In this paper, we present some properties of zero divisor graphs obtained from ring [Formula: see text], where [Formula: see text] and [Formula: see text] are primes. Also, we give some degree-based topological indices of this special graph.

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