Toroidality of Intersection Graphs of Ideals of Commutative Rings

Pucanović, Zoran S.; Petrović, Z

doi:10.1007/s00373-013-1292-1

Cited by 11 publications

(7 citation statements)

References 13 publications

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“…Graphs with genus 0 are planar and graphs of genus 1 are toroidal. Maimani et al [10], Pucanović [12] and Tamizh Chelvam et al [15] have studied about the genus of total graphs and other graphs associated with commutative rings. Let us first recall some known results connecting genus of graphs.…”

Section: Characterization Of Genus For Gt P (Z N )mentioning

confidence: 99%

Complement of the generalized total graph of Zn

Chelvam

Balamurugan

2019

Filomat

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Let R be a commutative ring with identity and H be a nonempty proper multiplicative prime subset of R. The generalized total graph of R is the (undirected) simple graph GT H (R) with all elements of R as the vertex set and two distinct vertices x and y are adjacent if and only if x + y ∈ H. The complement of the generalized total graph GT H (R) of R is the (undirected) simple graph with vertex set R and two distinct vertices x and y are adjacent if and only if x + y H. In this paper, we investigate certain domination properties of GT H (R). In particular, we obtain the domination number, independence number and a characterization for γ-sets in GT P (Z n) where P is a prime ideal of Z n. Further, we discuss properties like Eulerian, Hamiltonian, planarity, and toroidality of GT P (Z n).

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Section: Characterization Of Genus For Gt P (Z N )mentioning

confidence: 99%

Complement of the generalized total graph of Zn

Chelvam

Balamurugan

2019

Filomat

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“…A 5 has five copies of A 4 and any two A 4 in A 5 have non-trivial intersection, for otherwise |A 4 A 4 | = 144, which is not possible. Also H 1 := (12, 34) , H 2 := (12, 34), (13,24) , H 3 := (12, 34), (12354) , H 4 := (12, 34), (12453) , H 5 := (12, 34), (345) are proper subgroups of A 5 . Here H 1 is a subgroup of H i , for every i = 1, 2, 3, 4.…”

Section: Finite Non-solvable Groupsmentioning

confidence: 99%

“…Embeddability of graphs, associated with algebraic structures, on topological surfaces is considered in several recent papers [10,24,25,26]. Planarity of intersection graphs of subgroups finite groups were studied by Selçuk Kayacan et al in [22], and by H. Ahmedi et al in [15].…”

Section: Introductionmentioning

confidence: 99%

Toroidality and projective-planarity of intersection graphs of subgroups of finite groups

Rajkumar,

Devi

2015

Preprint

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Let G be a group. The intersection graph of subgroups of G, denoted by I (G), is a graph with all the proper subgroups of G as its vertices and two distinct vertices in I (G) are adjacent if and only if the corresponding subgroups having a non-trivial intersection in G. In this paper, we classify the finite groups whose intersection graph of subgroups are toroidal or projective-planar. In addition, we classify the finite groups whose intersection graph of subgroups are one of bipartite, complete bipartite, tree, star graph, unicyclic, acyclic, cycle, path or totally disconnected. Also we classify the finite groups whose intersection graph of subgroups does not contain one of K

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“…One of the most important topological properties of a graph is its genus, which can be orientable or non-orientable (crosscap). The genus of graphs associated with algebraic structures has been studied by many authors (see [12][13][14][15][16][17]). The planar zero-divisor graph was first explicitly characterized by Smith [18], and the characterization of commutative rings with projective zero-divisor graphs was obtained by Chiang-Hsieh [15].…”

Section: Introductionmentioning

confidence: 99%

Class of Crosscap Two Graphs Arising from Lattices–I

et al. 2023

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Let L be a lattice. The annihilating-ideal graph of L is a simple graph whose vertex set is the set of all nontrivial ideals of L and whose two distinct vertices I and J are adjacent if and only if I∧J=0. In this paper, crosscap two annihilating-ideal graphs of lattices with at most four atoms are characterized. These characterizations provide the classes of multipartite graphs, which are embedded in the Klein bottle.

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Toroidality of Intersection Graphs of Ideals of Commutative Rings

Cited by 11 publications

References 13 publications

Complement of the generalized total graph of Zn

Complement of the generalized total graph of Zn

Toroidality and projective-planarity of intersection graphs of subgroups of finite groups

Class of Crosscap Two Graphs Arising from Lattices–I

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