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Nilpotent groups whose difference graphs have positive genus

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Abstract

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, b are adjacent if there exists \(c \in G\) such that both a and b are powers of c. In this paper, we study the difference graph \(\mathcal {D}(G)\) 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 \(\mathcal {D}(G)\) is at most 2.

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Acknowledgements

We would like to thank the referee for his/her valuable suggestions which contributed to the final version of this paper.

Funding

The first author gratefully acknowledges for providing financial support to CSIR (09/719(0110)/2019-EMR-I) Government of India. The second author wishes to acknowledge the support of Core Research Grant (CRG/2022/001142) funded by SERB.

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

    Parveen & Jitender Kumar

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  1. Parveen
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  2. Jitender Kumar
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Correspondence to Jitender Kumar.

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Parveen, Kumar, J. Nilpotent groups whose difference graphs have positive genus. Ricerche mat (2023). https://doi.org/10.1007/s11587-023-00830-9

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