On the genus of dot product graph of a commutative ring

Selvakumar, K.; Ramanathan, Varun; Selvaraj, C.

doi:10.1007/s13226-022-00275-0

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2022

2024

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Cited by 3 publications

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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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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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Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

Mathil,

Baloda,

Kumar

et al. 2024

AKCE International Journal of Graphs and Combinatorics

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Characterization of rings with genus two prime ideal sum graphs

Mathil,

Kumar

2023

Asian-European J. Math.

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Let [Formula: see text] be a commutative ring with unity. The prime ideal sum graph of the ring [Formula: see text] is a simple undirected graph whose vertex set is the set of nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] is a prime ideal of [Formula: see text]. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus [Formula: see text].

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On the genus of dot product graph of a commutative ring

Cited by 3 publications

References 9 publications

On the cozero-divisor graphs associated to rings

On the cozero-divisor graphs associated to rings

Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

Characterization of rings with genus two prime ideal sum graphs

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