A Refinement of the Unit and Unitary Cayley Graphs of a Finite Ring

Naghipour, A. R.; Rezagholibeigi, Meysam

doi:10.4134/bkms.b150601

Cited by 4 publications

(1 citation statement)

References 17 publications

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“…There is no finite ring R such that the unit graph G(R) is projective. Some properties of the special case of the graph Γ(R, S) in the case that S = U (R) were studied by Naghipour et al in [17]. As a consequence of Corollary 4.11, we have the following corollary.…”

Section: γ(R S) With Nonorientable Genus Onementioning

confidence: 65%

On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring

Khorsandi,

Musawi

2020

Preprint

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Let R be a commutative ring and let U (R) be multiplicative group of unit elements of R. In 2012, Khashyarmanesh et al. defined generalized unit and unitary Cayley graph, Γ(R, G, S), corresponding to a multiplicative subgroup G of U (R) and a non-empty subset S of G with S −1 = {s −1 | s ∈ S} ⊆ S, as the graph with vertex set R and two distinct vertices x and y are adjacent if and only if there exists s ∈ S such that x+sy ∈ G. In this paper, we characterize all Artinian rings R whose Γ(R, U (R), S) is projective. This leads to determine all Artinian rings whose unit graphs, unitary Cayley garphs and co-maximal graphs are projective. Also, we prove that for an Artinian ring R whose Γ(R, U (R), S) has finite nonorientable genus, R must be a finite ring. Finally, it is proved that for a given positive integer k, the number of finite rings R whose Γ(R, U (R), S) has nonorientable genus k is finite.

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Section: γ(R S) With Nonorientable Genus Onementioning

confidence: 65%

On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring

Khorsandi,

Musawi

2020

Preprint

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Domination in generalized unit and unitary Cayley graphs of finite rings

Chelvam

Kathirvel

Balamurugan

2020

Indian J Pure Appl Math

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Generalized unit and unitary Cayley graphs of finite rings

Chelvam

Kathirvel

2019

J. Algebra Appl.

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Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

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A Refinement of the Unit and Unitary Cayley Graphs of a Finite Ring

Cited by 4 publications

References 17 publications

On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring

On the nonorientable genus of the generalized unit and unitary Cayley graphs of a commutative ring

Domination in generalized unit and unitary Cayley graphs of finite rings

Generalized unit and unitary Cayley graphs of finite rings

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