On a new extension of the zero-divisor graph (II)

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)).

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Zero Divisor Graph for the Ring of Eisenstein Integers Modulo n

Algebra ◽

10.1155/2014/146873 ◽

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.

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A generalized ideal-based zero-divisor graph

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500796 ◽

2015 ◽

Vol 14 (06) ◽

pp. 1550079 ◽

Cited By ~ 1

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.

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Zero-divisor Graph Generalizations

10.1007/978-3-030-88410-9_6 ◽

2021 ◽

pp. 239-292

Author(s):

David F. Anderson ◽

T. Asir ◽

Ayman Badawi ◽

T. Tamizh Chelvam

Keyword(s):

Zero Divisor ◽

Zero Divisor Graph

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On two extensions of the classical zero-divisor graph

São Paulo Journal of Mathematical Sciences ◽

10.1007/s40863-019-00155-2 ◽

2019 ◽

Vol 14 (1) ◽

pp. 341-350

Author(s):

Malik Bataineh ◽

Driss Bennis ◽

Jilali Mikram ◽

Fouad Taraza

Keyword(s):

Zero Divisor ◽

Zero Divisor Graph

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

Asian-European Journal of Mathematics ◽

10.1142/s179355712040001x ◽

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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