On the planarity, genus and crosscap of new extension of zero-divisor graph of commutative rings

Rehman, Nadeem ur; Nazim, Mohd; Selvakumar, K.

doi:10.1080/09728600.2022.2058437

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Historical Background and Main Results1

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2022

2024

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Cited by 6 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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