Classification of rings with toroidal Jacobson graph

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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“…More work related to the genus of graphs defined on rings can be found in [3,6,11,13,16,27,28,24,25,26,29,33,34].…”

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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“…Later Ma et al [5] studied these graphs and called them coprime graphs of groups. Further studies of these graphs were done in [3,6,9].…”

Section: Introductionmentioning

confidence: 99%

Classification of Groups according to the number of end vertices in the coprime graph

Alraqad¹,

Saeed²,

Alshawarbeh³

2018

Preprint

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In this paper we characterize groups according to the number of end vertices in the associated coprime graphs. An upper bound on the order of the group that depends on the number of end vertices is obtained. We also prove that 2−groups are the only groups whose coprime graphs have odd number of end vertices. Classifications of groups with small number of end vertices in the coprime graphs are given. One of the results shows that Z 4 and Z 2 × Z 2 are the only groups whose coprime graph has exactly three end vertices.

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

Selvakumar²,

Chelvam³

2022

Soft Comput

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Let R be a commutative ring with identity and let Ω(R) * be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph Γ r (R) of R is an undirected simple graph with Ω(R) * as the vertex set and two distinct vertices (x) and (y) in Ω(R) * are adjacent if and only if (x) (y) and (y) (x). In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one.

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Classification of rings with toroidal Jacobson graph

Abstract: Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Cited by 4 publications

References 19 publications

On the cozero-divisor graphs associated to rings

On the cozero-divisor graphs associated to rings

Classification of Groups according to the number of end vertices in the coprime graph

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

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