Some Properties of the Line Graphs Associated to the Total Graph of a Commutative Ring

Erić, Aleksandra; Pucanović, Zoran S.

doi:10.11648/j.pamj.20130202.11

Cited by 4 publications

(2 citation statements)

References 9 publications

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“…In 2010, Chiang-Hsieh et al [8] studied the genus of line graph of zero divisor graphs of rings and classified all finite commutative rings with genus at most two. In 2014, Eric et al [9], studied the girth and clique number of line graphs of total graphs of rings and determined when it is Eulerian.…”

Section: Introductionmentioning

confidence: 99%

Finite commutative rings whose line graphs of comaximal graphs have genus at most two

2024

Hacettepe Journal of Mathematics and Statistics

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Let $R$ be a ring with identity. The comaximal graph of $R$, denoted by $\Gamma(R)$, is a simple graph with vertex set $R$ and two different vertices $a$ and $b$ are adjacent if and only if $aR+bR=R$. Let $\Gamma_{2}(R)$ be a subgraph of $\Gamma(R)$ induced by $R\backslash\{U(R)\cup J(R)\}$. In this paper, we investigate the genus of the line graph $L(\Gamma(R))$ of $\Gamma(R)$ and the line graph $L(\Gamma_{2}(R))$ of $\Gamma_2(R)$. All finite commutative rings whose genus of $L(\Gamma(R))$ and $L(\Gamma_{2}(R))$ are 0, 1, 2 are completely characterized, respectively.

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Section: Introductionmentioning

confidence: 99%

Finite commutative rings whose line graphs of comaximal graphs have genus at most two

2024

Hacettepe Journal of Mathematics and Statistics

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“…If G is connected and if its line graph L(G) is known,one may,according to [6], completely determine G except in the case when L(G) is a triangle. Eric and Pucanovic [7] studied the structure and properties of the line graph associated to the total graph L(T Γ(R)). Knowing the structure of the total graph T Γ(M ), our goal is to investigate the structure of its line graph L(T Γ(M )) and to look into various relations between them.…”

Section: Introductionmentioning

confidence: 99%

On the Line Graph Associated to the Total Graph of a Module

Goswami

Saikia

2015

MATEMATIKA

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Let R be a commutative ring with unity and M be an R-module with T Γ(M ) be its total graph. The subject of this article is the investigation of the properties of the corresponding line graph L(T Γ(M )).In particular, we determine the girth and clique number of L(T Γ(M )). In addition to that, we find a condition for L(T Γ(M )) to be Eulerian.

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