On the Commuting Graph of Semidihedral Group

Kumar, Jitender; Dalal, S. S.; Baghel, Vedant

doi:10.1007/s40840-021-01111-0

Cited by 13 publications

(3 citation statements)

References 36 publications

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“…The semi-dihedral group, as a well-known non-abelian group, is a hot topic in group theory and has applications in graph theory [28] and symmetry classes of tensors [25,26]. In this paper, we consider the existence of PST on Cayley graphs over semi-dihedral groups.…”

Section: Introductionmentioning

confidence: 99%

Perfect quantum state transfer on Cayley graphs over semi-dihedral groups

Luo

Cao

Wang

et al. 2021

Linear and Multilinear Algebra

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Perfect quantum state transfer plays a crucial role in quantum information processing and quantum computation. There have been extensive study of perfect quantum state transfer on Cayley graphs over abelian groups. In this paper, we consider the existence of perfect quantum state transfer on Cayley graphs over semi-dihedral groups which are nonabelian groups. Using the representations of semi-dihedral groups, we provide some necessary and sufficient conditions for Cayley graphs over semi-dihedral groups admitting perfect quantum state transfer. By those conditions, we present examples of perfect quantum state transfer on Cayley graphs over semidihedral groups. In addition, we propose results about whether some new Cayley graphs over nonabelian groups admit perfect quantum state transfer.

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Section: Introductionmentioning

confidence: 99%

Perfect quantum state transfer on Cayley graphs over semi-dihedral groups

Luo

Cao

Wang

et al. 2021

Linear and Multilinear Algebra

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“…Afkhami et al [2] characterized all the finite groups whose commuting and noncommuting graphs are planar, projective planar and of genus one, respectively. Results on the commuting graph associated to groups can be found in [6,15,17,25,29,38] and references therein. Aalipour et al [1] characterize the finite group G such that the power graph P(G) and the commuting graph ∆(G) are not equal and hence they introduced a new graph between power graph and commuting graph, called enhanced power graph.…”

Section: Historical Background and Main Resultsmentioning

confidence: 99%

Nilpotent groups whose Difference graphs have positive genus

Parveen¹

2023

Preprint

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The power graph of a finite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is a simple undirected graph whose vertex set is the group G and two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we study the difference graph D(G) of a finite group G which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We characterize all the finite nilpotent groups G such that the genus (or cross-cap) of the difference graph D(G) is at most 2. 2020 Mathematics Subject Classification. 05C25. Key words and phrases. Enhanced power graph, power graph, nilpotent groups, genus and cross-cap of a graph.

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“…In the past decade, significant research has been directed at the matrices associated with graphs of groups, including the Laplacian spectrum and energy. These spectra have been studied by some researchers for the commuting graph of certain finite groups [6,7,8,9]. Comprehensive investigations of these spectra also covered for the power graph of some finite groups [10,11,12,13].…”

Section: Introductionmentioning

confidence: 99%

Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups

Mohamed,

Mohd Ali,

Bello

2024

Mal. J. Fund. Appl. Sci.

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This paper introduces cyclic order product prime graphs to examine the spectral graph properties of semi-dihedral groups, specifically focusing on the Laplacian spectrum and energy. Combining principles from cyclic graphs and order product prime graphs enhances understanding of group algebraic structures. In this context, the cyclic order product prime graph for a finite group is defined as one where two distinct vertices are adjacent if their generated subgroup is cyclic, and the product of their orders is a prime power. Our methodological approach begins with establishing a general presentation for these graphs within semi-dihedral groups. This foundational step is essential for deriving some properties, such as vertex degrees, the number of edges, and Laplacian characteristic polynomials. This information subsequently facilitates the determination of the Laplacian spectrum, characterized by seven eigenvalues of various multiplicities, and the computation of their Laplacian energy. The semi-dihedral group of order 16 is a particular example to illustrate the practicality and generality of our theorems.

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On the Commuting Graph of Semidihedral Group

Cited by 13 publications

References 36 publications

Perfect quantum state transfer on Cayley graphs over semi-dihedral groups

Perfect quantum state transfer on Cayley graphs over semi-dihedral groups

Nilpotent groups whose Difference graphs have positive genus

Laplacian Spectrum and Energy of the Cyclic Order Product Prime Graph of Semi-dihedral Groups

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