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On a Special Quotient of the Generating Graph of a Finite Group

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Abstract

Suppose G is a finite group with identity element 1. The generating graph \(\Gamma (G)\) is defined as a graph with vertex set G in such a way that two distinct vertices are connected by an edge if and only if they generate G and the Q-generating graph \(\Omega (G)\) is defined as the quotient graph \(\frac{\Gamma (G)\backslash \{ 1\}}{\mathcal {C}^\star (G)}\), where \(\mathcal {C}^\star (G)\) is the set of all non-identity conjugacy classes of G and \(\Gamma (G)\backslash \{ 1\}\) is a graph obtained from \(\Gamma (G)\) by removing the vertex 1. In this paper, some structural properties of this graph are investigated. The structure of Q-generating graphs of dihedral, semidihedral, dicyclic and all sporadic groups other than M, B and \(Fi_{24}^\prime \) is also presented.

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Acknowledgements

We are indebted to the anonymous referee for his/her suggestions and helpful remarks that leaded us to rearrange the paper. We are very grateful to Professor Martin Isaacs for his interesting comments and for giving us the corrected proof of Theorem 3.1. This research is partially supported by the University of Kashan under Grant Number 364988/38.

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  1. Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, 87317-53153, Islamic Republic of Iran

    A. R. Ashrafi, F. Gholaminezhad & M. Yousefi

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  1. A. R. Ashrafi
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  2. F. Gholaminezhad
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  3. M. Yousefi
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Correspondence to A. R. Ashrafi.

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Communicated by Rosihan M. Ali.

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Ashrafi, A.R., Gholaminezhad, F. & Yousefi, M. On a Special Quotient of the Generating Graph of a Finite Group. Bull. Malays. Math. Sci. Soc. 42, 1831–1852 (2019). https://doi.org/10.1007/s40840-017-0574-9

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