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On the Commuting Graph of Semidihedral Group

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Abstract

The commuting graph \(\varDelta (G)\) of a finite non-abelian group G is a simple graph with vertex set G, and two distinct vertices x, y are adjacent if \(xy = yx\). In this paper, first we discuss some properties of \(\varDelta (G)\). We determine the edge connectivity and the minimum degree of \(\varDelta (G)\) and prove that both are equal. Then, other graph invariants, namely: matching number, clique number, boundary vertex, of \(\varDelta (G)\) are studied. Also, we give necessary and sufficient condition on the group G such that the interior and center of \(\varDelta (G)\) are equal. Further, we investigate the commuting graph of the semidihedral group \(SD_{8n}\). In this connection, we discuss various graph invariants of \(\varDelta (SD_{8n})\) including vertex connectivity, independence number, matching number and detour properties. We also obtain the Laplacian spectrum, metric dimension and resolving polynomial of \(\varDelta (SD_{8n})\).

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Acknowledgements

The authors are very grateful to the referees for their valuable suggestions which lead to an improvement of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology and Science Pilani, Pilani, 333031, India

    Jitender Kumar, Sandeep Dalal & Vedant Baghel

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  1. Jitender Kumar
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  2. Sandeep Dalal
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  3. Vedant Baghel
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Correspondence to Jitender Kumar.

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Communicated by Rosihan M. Ali.

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Jitender Kumar: Supported by “Science and Engineering Research Board of India under MATRICS Grant (MTR/2018/000779)”.

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Kumar, J., Dalal, S. & Baghel, V. On the Commuting Graph of Semidihedral Group. Bull. Malays. Math. Sci. Soc. 44, 3319–3344 (2021). https://doi.org/10.1007/s40840-021-01111-0

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  • DOI: https://doi.org/10.1007/s40840-021-01111-0

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