Abstract
The enhanced power graph of a finite group \(G\), denoted by \(P_E(G)\), is a simple undirected graph whose vertex set is G and two distinct vertices x, y are adjacent if \(x, y \in \langle z \rangle\) for some \(z \in G\). In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of \(P_E(G)\) are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain the upper and lower bounds of the Wiener index of \(P_E(G)\), where G is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Aalipour, S. Akbari, P. J. Cameron, R. Nikandish and F. Shaveisi, On the structure of the power graph and the enhanced power graph of a group, Electron. J. Combin., 24 (2017), Paper No. 3.16, 18 pp.
M. Aschbacher, Finite Group Theory, Cambridge University Press (Cambridge, 2000).
S. Bera and A. K. Bhuniya, On enhanced power graphs of finite groups, J. Algebra Appl., 17 (2018), 1850146.
S. Bera, H. K. Dey and S. K. Mukherjee, On the connectivity of enhanced power graphs of finite groups, Graphs Combin., 37 (2021), 591–603.
I. Chakrabarty, S. Ghosh and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum, 78 (2009), 410–426.
S. Chattopadhyay, K. L. Patra and B. K. Sahoo, Minimal cut-sets in the power graphs of certain finite non-cyclic groups, Comm. Algebra, 49 (2021), 1195–1211.
S. Dalal and J. Kumar, On enhanced power graphs of certain groups, Discrete Math. Algorithms Appl., 13 (2021), 2050099.
D. S. Dummit and R. M. Foote, Abstract Algebra, Prentice Hall, Inc. (Englewood Cliffs, NJ, 1991).
L. A. Dupont, D. G. Mendoza and M. Rodr´ıguez, The enhanced quotient graph of the quotient of a finite group, arXiv:1707.01127 (2017).
L. A. Dupont, D. G. Mendoza and M. Rodr´ıguez, The rainbow connection number of enhanced power graph, arXiv:1708.07598 (2017).
A. Hamzeh and A. R. Ashrafi, Automorphism groups of supergraphs of the power graph of a finite group, European J. Combin., 60 (2017), 82–88.
U. Hayat, M. Umer, I. Gutman, B. Davvaz and A. Nolla de Celis, A novel method to construct NSSD molecular graphs, Open Math., 17 (2019), 1526–1537.
A. V. Kelarev, On undirected Cayley graphs, Australas. J. Combin., 25 (2002), 73–78.
A. V. Kelarev, Ring Constructions and Applications, World Scientific Publishing Co., Inc. (River Edge, NJ, 2002).
A. Kelarev, Graph Algebras and Automata, Marcel Dekker, Inc. (New York, 2003).
A. V. Kelarev, Labelled Cayley graphs and minimal automata, Australas. J. Combin., 30 (2004), 95–101.
A. Kelarev, J. Ryan, and J. Yearwood, Cayley graphs as classifiers for data mining: the influence of asymmetries, Discrete Math., 309 (2009), 5360–5369.
X. Ma, A. Doostabadi and K. Wang, Notes on the diameter of the complement of the power graph of a finite group, arXiv:2112.13499v2 (2021).
X. Ma, R. Fu, X. Lu, M. Guo and Z. Zhao, Perfect codes in power graphs of finite groups, Open Math., 15 (2017), 1440–1449.
X. Ma, A. Kelarev, Y. Lin and K. Wang, A survey on enhanced power graphs of finite groups, Electron. J. Graph Theory Appl., 10 (2022), 89–111.
X. Ma and Y. She, The metric dimension of the enhanced power graph of a finite group, J. Algebra Appl., 19 (2020), 2050020, 14 pp.
A. R. Moghaddamfar, S. Rahbariyan and W. J. Shi, Certain properties of the power graph associated with a finite group, J. Algebra Appl., 13 (2014), 1450040.
Parveen and J. Kumar, The complement of enhanced power graph of a finite group, arXiv:2207.04641 (2022).
R. Prasad Panda, S. Dalal and J. Kumar, On the enhanced power graph of a finite group, Comm. Algebra, 49 (2021), 1697–1716.
Y. Segev, On finite homomorphic images of the multiplicative group of a division algebra, Ann. of Math. (2), 149 (1999), 219–251.
Y. Segev, The commuting graph of minimal nonsolvable groups, Geom. Dedicata, 88 (2001), 55–66.
Y. Segev and G. M. Seitz, Anisotropic groups of type \(A_n\) and the commuting graph of finite simple groups, Pacific J. Math., 202 (2002), 125–225.
D. B. West, Introduction to Graph Theory, Prentice Hall, Inc. (Upper Saddle River, NJ, 1996).
S. Zahirovi´c, I. Boˇsnjak and R. Madarász, A study of enhanced power graphs of finite groups, J. Algebra Appl., 19 (2020), 2050062.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author wishes to acknowledge the support of Core Research Grant (CRG/2022/001142) funded by SERB.
The third author gratefully acknowledge for providing financial support to CSIR (09/719(0110)/2019-EMR-I) government of India.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kumar, J., Ma, X., Parveen et al. Certain properties of the enhanced power graph associated with a finite group. Acta Math. Hungar. 169, 238–251 (2023). https://doi.org/10.1007/s10474-023-01304-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-023-01304-y