Abstract
Let G be a group with identity element e. The proper power graph and proper enhanced power graph of G, are denoted by \(\Gamma ^*_{P}(G)\) and \(\Gamma ^*_{EP}(G)\), respectively. Also, the prime graph of G is denoted by \(\Gamma _{GK}(G)\). In an article, Aalipour et al. (Electronic J. Combin. 24(3) (2017) 3–16) asked which groups do have the property that \(\Gamma ^*_{P}(G)\) is connected? In this paper, we show that if \(\Gamma _{GK}(G)\) is disconnected, then \(\Gamma ^*_{P}(G)\) and \(\Gamma ^*_{EP}(G)\) are disconnected. Moreover, we prove that if G is a nilpotent group which is not a p-group, then \(\Gamma ^*_{EP}(G)\) is a connected graph.
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The authors are grateful to the referee for helpful remarks.
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Communicated by Manoj Kumar Yadav.
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Babai, A., Mahmoudifar, A. On a problem about the connectivity of the proper enhanced power graph of a finite group. Proc Math Sci 132, 28 (2022). https://doi.org/10.1007/s12044-022-00665-8
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DOI: https://doi.org/10.1007/s12044-022-00665-8