The unit graph of a left Artinian ring

Heydari, F.; Nikmehr, M. J.

doi:10.1007/s10474-012-0250-3

Cited by 18 publications

(12 citation statements)

References 8 publications

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“…McKenzie and Schneider. However, Anderson and Badawi [3] proved that for each integer n ≥ 1, there exists a ring R such that its total graph has diameter n. For the unit graph, Heydari and Nikmehr [12] proved that its diameter is 1, 2, 3 or ∞ for an Artinian ring.…”

Section: Huadong Su and Yangjiang Weimentioning

confidence: 99%

See 1 more Smart Citation

The Diameter of Unit Graphs of Rings

Su¹,

Wei²

2019

Taiwanese J. Math.

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Let R be a ring. The unit graph of R, denoted by G(R), is the simple graph defined on all elements of R, and where two distinct vertices x and y are linked by an edge if and only if x + y is a unit of R. The diameter of a simple graph G, denoted by diam(G), is the longest distance between all pairs of vertices of the graph G. In the present paper, we prove that for each integer n ≥ 1, there exists a ring R such that n ≤ diam(G(R)) ≤ 2n. We also show that diam(G(R)) ∈ {1, 2, 3, ∞} for a ring R with R/J(R) self-injective and classify all those rings with diam(G(R)) = 1, 2, 3 and ∞, respectively. This extends [12, Theorem 2 and Corollary 1].

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Section: Huadong Su and Yangjiang Weimentioning

confidence: 99%

“…Maimani et al gave the necessary and sufficient conditions for unit graphs to be Hamiltonian in [15]. Heydari and Nikmehr investigated the unit graph of a left Artinian ring in [12]. Afkhami and Khosh-Ahang studied the unit graphs of rings of polynomials and power series in [1].…”

Section: Huadong Su and Yangjiang Weimentioning

confidence: 99%

The Diameter of Unit Graphs of Rings

Su¹,

Wei²

2019

Taiwanese J. Math.

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“…There are many papers on assigning a graph to rings, see for example [1], [2], [6], [7] and [10]. When one assigns a graph to an algebraic structure numerous interesting algebraic problems arise from the translation of some graph-theoretic parameters such as clique number, chromatic number, independence number and so on.…”

Section: Introductionmentioning

confidence: 99%

The Annihilator Ideal Graph of a Commutative Ring

Alibemani¹,

Bakhtyiari²,

Nikandish³

et al. 2015

Journal of the Korean Mathematical Society

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Abstract. Let R be a commutative ring with unity. The annihilator ideal graph of R, denoted by Γ Ann (R), is a graph whose vertices are all non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if I ∩ Ann(J) = {0} or J ∩ Ann(I) = {0}. In this paper, we study some connections between the graph-theoretic properties of this graph and some algebraic properties of rings. We characterize all rings whose annihilator ideal graphs are totally disconnected. Also, we study diameter, girth, clique number and chromatic number of this graph. Moreover, we study some relations between annihilator ideal graph and zero-divisor graph associated with R. Among other results, it is proved that for a Noetherian ring R if Γ Ann (R) is triangle free, then R is Gorenstein.

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“…There are many papers on assigning a graph to a ring R, see [1][2][3][4]9,10]. In this paper, we introduce the M-principal graph of R, denoted by M − PG(R), where M is an R-module.…”

Section: Introductionmentioning

confidence: 99%

The M-principal graph of a commutative ring

Nikmehr

Heydari

2014

Period Math Hung

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Let R be a commutative ring and M be an R-module. In this paper, we introduce the M-principal graph of R, denoted by M − PG(R). It is the graph whose vertex set is R\{0}, and two distinct vertices x and y are adjacent if and only if x M = y M. In the special case that M = R, M − PG(R) is denoted by PG(R). The basic properties and possible structures of these two graphs are studied. Also, some relations between PG(R) and M − PG(R) are established.

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The unit graph of a left Artinian ring

Cited by 18 publications

References 8 publications

The Diameter of Unit Graphs of Rings

The Diameter of Unit Graphs of Rings

The Annihilator Ideal Graph of a Commutative Ring

The M-principal graph of a commutative ring

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