Characterization of rings with genus two cozero-divisor graphs

Praveen, Mathil,; Baloda, Barkha; Kumar, Jitendra

doi:10.48550/arxiv.2212.12900

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2022

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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%