Chordality of graphs associated to commutative rings

Nikseresht, Ashkan

doi:10.3906/mat-1801-71

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…The other graph properties such as the well-coveredness, Hamiltonicity and chordality of the unit graphs of rings were examined in [250][251][252], respectively. In [251], a necessary and sufficient condition for the unit graph of a finite commutative ring to be Hamiltonian was derived, by constructing a graph based on the structural properties of the rings, whose unit graph was connected as obtained in [225].…”

Section: Unit Graph Of a Ringmentioning

confidence: 99%

“…Followeing the study on Hamiltonicity, the chordality in the unit graphs of finite commutative rings was studied in [252], where the rings having quotient ring R J R as a product of fields were characterised based on the chordality of the unit graphs, and, in [250], a necessary and sufficient condition under which the unit graphs of finite commutative rings were well-covered was deduced, from which the unit graphs whose edge rings were Cohen-Macaulay and Gorenstein were characterised, as given in Theorem 113. This characterisation led to the identification of a large class of non-Cohen-Macaulay graphs.…”

Section: Unit Graph Of a Ringmentioning

confidence: 99%

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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: Unit Graph Of a Ringmentioning

confidence: 99%

Section: Unit Graph Of a Ringmentioning

confidence: 99%

Graphs Defined on Rings: A Review

Madhumitha,

Naduvath

2023

Mathematics

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“…For instance, it is known that the class of chordal graphs is perfect; see Dirac [17]. The notion of perfectness, weakly perfectness and chordalness of graphs associated with algebraic structures has been an active area of research; see [1], [5], [7], [8], [15], [38], [39], [42], etc.…”

Section: Introductionmentioning

confidence: 99%

Coloring of zero-divisor graphs of posets and applications to graphs associated with algebraic structures

Khandekar¹,

Joshi²

2022

Preprint

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In this paper, we characterize chordal and perfect zero-divisor graphs of finite posets. Also, it is proved that the zero-divisor graphs of finite posets and the complement of zero-divisor graphs of finite 0-distributive posets satisfy the Total Coloring Conjecture. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graph of rings, the annihilating ideal graphs, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of cyclic groups. In fact, it is proved that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

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Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures

Khandekar,

Joshi

2023

Mathematica Slovaca

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In this paper, we characterize the perfect zero-divisor graphs and chordal zero-divisor graphs (its complement) of ordered sets. These results are applied to the zero-divisor graphs of finite reduced rings, the comaximal ideal graphs of rings, the annihilating ideal graphs of rings, the intersection graphs of ideals of rings, and the intersection graphs of subgroups of groups. In fact, it is shown that these graphs associated with a commutative ring R with identity can be effectively studied via the zero-divisor graph of a specially constructed poset from R.

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Chordality of graphs associated to commutative rings

Cited by 3 publications

References 15 publications

Graphs Defined on Rings: A Review

Graphs Defined on Rings: A Review

Coloring of zero-divisor graphs of posets and applications to graphs associated with algebraic structures

Chordal and Perfect Zero-Divisor Graphs of Posets and Applications to Graphs Associated with Algebraic Structures

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