On the genus of reduced cozero-divisor graph of commutative rings

E, Jesili; Selvakumar, K.; Chelvam, T. Tamizh

doi:10.1007/s00500-022-07616-5

Cited by 5 publications

(2 citation statements)

References 18 publications

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“…Further, Pucanović et al [22] classified the commutative rings for which the intersection graph of ideals has genus two. All the rings with genus one (or two) reduced cozero-divisor graphs have been classified in [12,17].…”

Section: Historical Background and Main Resultsmentioning

confidence: 99%

On the cozero-divisor graphs associated to rings

Praveen

Baloda

Kumar

2022

AKCE International Journal of Graphs and Combinatorics

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Let R be a ring with unity. The idempotent graph G Id (R) of a ring R is an undirected simple graph whose vertices are the set of all the elements of ring R and two vertices x and y are adjacent if and only if x + y is an idempotent element of R. In this paper, we obtain a necessary and sufficient condition on the ring R such that G Id (R) is planar. We prove that G Id (R) cannot be an outerplanar graph. Moreover, we classify all the finite non-local commutative rings R such that G Id (R) is a cograph, split graph and threshold graph, respectively. We conclude that latter two graph classes of G Id (R) are equivalent if and only2010 Mathematics Subject Classification. 05C25.

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Section: Historical Background and Main Resultsmentioning

confidence: 99%

On the cozero-divisor graphs associated to rings

Praveen

Baloda

Kumar

2022

AKCE International Journal of Graphs and Combinatorics

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“…All the non-local commutative rings whose zero-divisor graphs having genus either 1 or 2, have been classified in [7,33]. Jesili et al [16] characterized all the commutative non-local rings whose reduced cozero-divisor graph has genus at most one. Biswas et al [12] provided all the finite commutative rings whose generalised co-maximal graph has genus at most two.…”

Section: Introductionmentioning

confidence: 99%

A study of upper ideal relation graphs of rings

Baloda,

Mathil,

Kumar

2023

AKCE International Journal of Graphs and Combinatorics

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Let R be a ring with unity. The upper ideal relation graph Γ U (R) of the ring R is a simple undirected graph whose vertex set is the set of all non-unit elements of R and two distinct vertices x, y are adjacent if and only if there exists a non-unit element z ∈ R such that the ideals (x) and (y) contained in the ideal (z). In this article, we classify all the non-local finite commutative rings whose upper ideal relation graphs are split graphs, threshold graphs and cographs, respectively. In order to study topological properties of Γ U (R), we determine all the non-local finite commutative rings R whose upper ideal relation graph has genus at most 2. Further, we precisely characterize all the non-local finite commutative rings for which the crosscap of Γ U (R) is either 1 or 2. 2020 Mathematics Subject Classification. 05C25.

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Upper ideal relation graphs associated to rings

Baloda,

Kumar

2024

Soft Comput

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On the genus of reduced cozero-divisor graph of commutative rings

Cited by 5 publications

References 18 publications

On the cozero-divisor graphs associated to rings

On the cozero-divisor graphs associated to rings

A study of upper ideal relation graphs of rings

Upper ideal relation graphs associated to rings

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