On Domination in Graphs from Commutative Rings: A Survey

Chelvam, T. Tamizh; Asir, T.; Selvakumar, K.

doi:10.1007/978-981-10-1651-6_23

Cited by 3 publications

(2 citation statements)

References 19 publications

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“…Furthermore, review papers that focus on a particular property of graphs defined on rings can also be found in the literature (c.f. [26][27][28]), whereas there was no comprehensive review found related to the variants of Cayley graphs defined on rings. This motivated us to create a literature hub for these graphs defined on common grounds and to systematically analyse the studies that have been carried out on these graphs to understand the pattern and dynamics of research in this area.…”

Section: Introductionmentioning

confidence: 99%

Graphs Defined on Rings: A Review

Madhumitha,

Naduvath

2023

Mathematics

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The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area.

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Section: Introductionmentioning

confidence: 99%

Graphs Defined on Rings: A Review

Madhumitha,

Naduvath

2023

Mathematics

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“…Throughout this paper, R denotes a commutative ring without identity. Let ZðRÞ; NilðRÞ; and ann R ðSÞ denotes the set of all zero-divisors of R, the set of all nilpotent elements of R and the set of all annihilators of a subset S of R, respectively, and let R Ã ¼ Rnf0g: Several authors [1,2,[20][21][22] studied about graphs from algebraic structures and in particular graphs from rings. The study of graphs from commutative rings was initiated by Beck [11].…”

Section: Introductionmentioning

confidence: 99%

Toroidal zero-divisor graphs of decomposable commutative rings without identity

Kalaimurugan

Vignesh

Chelvam

2020

Bol. Soc. Mat. Mex.

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Let R be a commutative ring without identity. The zero-divisor graph of R, denoted by CðRÞ; is a graph with vertex set ZðRÞnf0g; which is the set of all non-zero zerodivisor elements of R and two vertices x and y are adjacent if and only if xy ¼ 0: In this paper, we characterize (up to isomorphism) all finite decomposable commutative rings without identity whose zero-divisor graphs are toroidal.

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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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On Domination in Graphs from Commutative Rings: A Survey

Cited by 3 publications

References 19 publications

Graphs Defined on Rings: A Review

Graphs Defined on Rings: A Review

Toroidal zero-divisor graphs of decomposable commutative rings without identity

Generalized unit and unitary Cayley graphs of finite rings

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