On the Commuting Graph of Dihedral Group

Ali, Faisal; Salman, Muhammad; Huang, Shuliang

doi:10.1080/00927872.2015.1053488

Cited by 41 publications

(36 citation statements)

References 11 publications

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“…(i) e commuting graph of a nonabelian group Γ is denoted by Γ G � C(Γ, Ω) with vertex set Ω ⊆ Γ. For two distinct elements x, y ∈ Ω, x ∼ y in Γ G (x and y form an edge in Γ G ) if and only if xy � yx in Γ. e concept of commuting graphs on noncentral elements of a group has been studied by various researchers (see [16,17]). (ii) e noncommuting graph of a nonabelian group G Γ is a graph with vertex set…”

Section: Introductionmentioning

confidence: 99%

Some Topological Invariants of Graphs Associated with the Group of Symmetries

Wei

Salman

Ali

et al. 2020

Journal of Chemistry

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A topological index is a quantity that is somehow calculated from a graph (molecular structure), which reflects relevant structural features of the underlying molecule. It is, in fact, a numerical value associated with the chemical constitution for the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. A large number of properties like physicochemical properties, thermodynamic properties, chemical activity, and biological activity can be determined with the help of various topological indices such as atom-bond connectivity indices, Randić index, and geometric arithmetic indices. In this paper, we investigate topological properties of two graphs (commuting and noncommuting) associated with an algebraic structure by determining their Randić index, geometric arithmetic indices, atomic bond connectivity indices, harmonic index, Wiener index, reciprocal complementary Wiener index, Schultz molecular topological index, and Harary index.

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Section: Introductionmentioning

confidence: 99%

Some Topological Invariants of Graphs Associated with the Group of Symmetries

Wei

Salman

Ali

et al. 2020

Journal of Chemistry

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“…Recently, there was much work done in computing the metric dimension of graphs associated with algebraic structures. Calculating the metric dimension for the commuting graph of a dihedral group was done in [1], for the zero-divisor graphs of commutative rings in [9,10,12], for the compressed zero-divisor graphs of commutative rings in [13], for total graphs of finite commutative rings in [6], for some graphs of modules in [11] and for annihilator graphs of commutative rings in [15]. Motivated by these papers, we study the metric dimension of another graph associated with a commutative ring.…”

Section: Introductionmentioning

confidence: 99%

On the metric dimension of strongly annihilating-ideal graphs of commutative rings

Soleymanivarniab

Nikandish

Tehranian

2020

Acta Universitatis Sapientiae, Mathematica

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Let 𝒭 be a commutative ring with identity and 𝒜(𝒭) be the set of ideals with non-zero annihilator. The strongly annihilating-ideal graph of 𝒭 is defined as the graph SAG(𝒭) with the vertex set 𝒜 (𝒭)* = 𝒜 (𝒭) \{0} and two distinct vertices I and J are adjacent if and only if I ∩ Ann(J) ≠ (0) and J ∩ Ann(I) ≠ (0). In this paper, we study the metric dimension of SAG(𝒭) and some metric dimension formulae for strongly annihilating-ideal graphs are given.

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“…The study of algebraic structures, using the properties of graphs, has become an exciting research topic in these years, leading to many fascinating results and raising questions. The commuting graph of a group is studied by various author (see [1]- [3] and [6]- [10]). In [3], the certain properties of the commuting graph of dihedral group are studied.…”

Section: Introductionmentioning

confidence: 99%

“…The commuting graph of a group is studied by various author (see [1]- [3] and [6]- [10]). In [3], the certain properties of the commuting graph of dihedral group are studied. In this paper, we have studied those properties for the commuting graph of generalized dihedral group…”

Section: Introductionmentioning

confidence: 99%