On the Total Graph of a Finite Commutative Ring

Shekarriz, Mohammad Hadi; Haghighi, Mohammad Hassan Shirdareh; Sharif, Habib

doi:10.1080/00927872.2011.585680

Cited by 21 publications

(12 citation statements)

References 5 publications

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“…As a consequence of Proposition 2.3 we obtain the following result, see also [7,Theorem 4.1]. We now proceed to investigate the total graphs of arbitrary finite rings.…”

Section: The Domination Numbermentioning

confidence: 75%

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The Total Graphs of Finite Rings

Dolžan

Oblak

2015

Communications in Algebra

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“…As a consequence of Proposition 2.3 we obtain the following result, see also [7,Theorem 4.1]. We now proceed to investigate the total graphs of arbitrary finite rings.…”

Section: The Domination Numbermentioning

confidence: 75%

“…Some properties of the total graph over a non-commutative finite ring can be proved by the same arguments as in the commutative case. For example, the arguments in the proofs of Theorems 2.7 and 3.3 from [7] and Theorem 3.3 and Theorem 3.4 from [2] are valid also in the non-commutative case, see Lemmas 2.4, 2.5 and 2.6.…”

Section: Preliminariesmentioning

confidence: 99%

The Total Graphs of Finite Rings

Dolžan

Oblak

2015

Communications in Algebra

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“…Proof. Since every finite ring decomposes into a product of finite local rings (see [6,Theorem 8.7]), by Part (b) of [21,Theorem 5.2], we obtain that T (Γ(R)) ∼ = CAY(R). Now, the assertion follows from Theorem 16.…”

Section: The Connectivity Of the Cayley Graph Of A Ringmentioning

confidence: 99%

On the Cayley Graph of a Commutative Ring with Respect to its Zero-divisors

Aalipour

Akbari

2016

Communications in Algebra

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Let R be a commutative ring with unity and R + and Z * (R) be the additive group and the set of all non-zero zero-divisors of R, respectively. We denote by CAY(R) the Cayley graph Cay(R + , Z * (R)). In this paper, we study CAY(R). Among other results, it is shown that for every zero-dimensional non-local ring R, CAY(R) is a connected graph of diameter 2. Moreover, for a finite ring R, we obtain the vertex connectivity and the edge connectivity of CAY(R). We investigate rings R with perfect CAY(R) as well. We also study Reg(CAY(R)) the induced subgraph on the regular elements of R. This graph gives a family of vertex transitive graphs. We show that if R is a Noetherian ring and Reg(CAY(R)) has no infinite clique, then R is finite. Furthermore, for every finite ring R, the clique number and the chromatic number of Reg(CAY(R)) are determined. * Proof. Parts (i) and (iii) are obvious. Part (ii) follows from Z(R) = m. Part (iv) holds for every Cayley graph of a group. To prove the last part, note that under an automorphism of graph G, any component of G is isomorphically mapped to another component. Since CAY(R) is vertex-transitive, we conclude that the components of CAY(R) are isomorphic and so (v) is proved.Remark 2. CAY(R) is vertex transitive but it is not necessarily edge transitive. To see this, consider CAY(Z 6 ) ∼ = K 2 K 3 which is not edge transitive (see Figure 1).

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“…The radius of T (Γ(R)) was computed in [13]. The domination number of T (Γ(R)) is determined independently in both [7] and [16]. For a nite commutative ring R, a characterization of Eulerian T (Γ(R)) is given in [16].…”

Section: Introductionmentioning

confidence: 99%

“…The domination number of T (Γ(R)) is determined independently in both [7] and [16]. For a nite commutative ring R, a characterization of Eulerian T (Γ(R)) is given in [16]. Minimum zero -sum k-ows for T (Γ(R)) are considered in [15].…”

Section: Introductionmentioning

confidence: 99%

Some Properties of the Total Graph and Regular Graph of a Commutative Ring

Ghanem¹,

Nazzal²

2017

HJMS

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Let R be a commutative ring with unity. The total graph of R, T (Γ(R)), is the simple graph with vertex set R and two distinct vertices are adjacent if their sum is a zero-divisor in R. Let Reg(Γ(R)) and Z(Γ(R)) be the subgraphs of T (Γ(R)) induced by the set of all regular elements and the set of zero-divisors in R, respectively. We determine when each of the graphs T (Γ(R)), Reg(Γ(R)), and Z(Γ(R)) is locally connected, and when it is locally homogeneous. When each of Reg(Γ(R)) and Z(Γ(R)) is regular and when it is Eulerian.

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On the Total Graph of a Finite Commutative Ring

Cited by 21 publications

References 5 publications

The Total Graphs of Finite Rings

The Total Graphs of Finite Rings

On the Cayley Graph of a Commutative Ring with Respect to its Zero-divisors

Some Properties of the Total Graph and Regular Graph of a Commutative Ring

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