On the Genus of the Intersection Graph of Ideals of a Commutative Ring

Pucanović, Zoran S.; Radovanović, Marko; Erić, Aleksandra

doi:10.1142/s0219498813501557

Cited by 15 publications

(6 citation statements)

References 24 publications

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“…Moreover, an improvement over the existing results concerning the planarity of intersection graphs of rings is also presented. 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%

On the cozero-divisor graphs associated to rings

Praveen

Baloda

Kumar

2022

AKCE International Journal of Graphs and Combinatorics

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“…Planar and toroidal graphs that are intersection ideal graphs of Artinian commutative rings are classified in [24]. Pucanović et al [25] characterized all the graph classes of genus 2 that are intersection graphs of ideals of some commutative rings. The intersection ideal graphs with crosscap at most two of all the Artinian commutative rings have been investigated by Ramanathan [26].…”

Section: Introductionmentioning

confidence: 99%

A study of upper ideal relation graphs of rings

Baloda,

Mathil,

Kumar

2023

AKCE International Journal of Graphs and Combinatorics

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Let R be a ring with unity. The upper ideal relation graph Γ U (R) of the ring R is a simple undirected graph whose vertex set is the set of all non-unit elements of R and two distinct vertices x, y are adjacent if and only if there exists a non-unit element z ∈ R such that the ideals (x) and (y) contained in the ideal (z). In this article, we classify all the non-local finite commutative rings whose upper ideal relation graphs are split graphs, threshold graphs and cographs, respectively. In order to study topological properties of Γ U (R), we determine all the non-local finite commutative rings R whose upper ideal relation graph has genus at most 2. Further, we precisely characterize all the non-local finite commutative rings for which the crosscap of Γ U (R) is either 1 or 2. 2020 Mathematics Subject Classification. 05C25.

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“…The intersection graphs of ideals of direct product of rings have been discussed in [19]. Pucanovic et al [28] classified all graphs of genus two that are intersection graphs of ideals of some commutative rings and obtain some lower bounds for the genus of the intersection graph of ideals of a non local commutative ring. In [13], Das characterized the positive integer n for which the intersection graph of ideals of Z n is perfect.…”

Section: Introductionmentioning

confidence: 99%

On the intersection ideal graph of semigroups

Baloda¹,

Kumar²

2022

Preprint

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The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial.In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected then diam(Γ(S)) ≤ 2.Further we classify the semigroups such that the diameter of their intersection graph is two. Other graph invariants, namely perfectness, planarity, girth, dominance number, clique number, independence number etc. are also discussed.Finally, if S is union of n minimal left ideals then we obtain the automorphism group of Γ(S).Recently, Chakrabarty et al. [8] introduced the notion of intersection ideal graph of rings. The intersection ideal graph Γ(R) of a ring R is an undirected simple graph whose vertex set is the collection of nontrivial left ideals of R and two distinct vertices I, J are adjacent if and only if I ∩ J = {0}. They characterized the rings R for which the graph Γ(R) is connected and obtain several necessary and sufficient conditions on a ring R such that Γ(R) is complete. Planarity of intersection graphs of ideals of ring with unity is described in [15] and domination number in 2010 Mathematics Subject Classification. 05C25.

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On the Genus of the Intersection Graph of Ideals of a Commutative Ring

Cited by 15 publications

References 24 publications

On the cozero-divisor graphs associated to rings

On the cozero-divisor graphs associated to rings

A study of upper ideal relation graphs of rings

On the intersection ideal graph of semigroups

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