4,648
Views
7
CrossRef citations to date
0
Altmetric
Articles

Recent developments on the power graph of finite groups – a survey

ORCID Icon, ORCID Icon, ORCID Icon & ORCID Icon
Pages 65-94 | Received 19 May 2021, Accepted 03 Jul 2021, Published online: 26 Jul 2021

References

  • Aalipour, G., Akbari, S., Cameron, P. J., Nikandish, R, Shaveisi, F. (2017). On the structure of the power graph and the enhanced power graph of a group. Electron. J. Comb. 24(3): P3.16.
  • Abawajy, J., Kelarev, A., Chowdhury, M. (2013). Power graphs: A survey. Ejgta. 1(2): 125–147.
  • Akbari, N., Ashrafi, A. R. (2015). Note on the power graph of finite simple groups. Quasigroups Relat. Syst 23(2): 165–173.
  • Alireza, D., Ahmad, E, Abbas, J. (2015). Some results on the power graphs of finite groups. Sci. Asia 41(1): 73–78.
  • Amiri, H., Jafarian Amiri, S. M., Isaacs, I. M. (2009). Sums of element orders in finite groups. Comm. Algebra 37(9): 2978–2980.
  • Ashraf, A. R., Gholami, A., Mehranian, Z. (2017). Automorphism group of certain power graphs of finite groups. Ejgta. 5(1): 70–82.
  • Bhuniya, A. K., Kumar Mukherjee, S. (2017). On the power graph of the direct product of two groups, Electron Notes Discrete Math. 63: 197–202.
  • Bollobás, B. (2013). Modern Graph Theory, Vol. 184. New York: Springer-Verlag.
  • Bondy, J. A., Murty, U. S. R. (2008). Graph Theory with Applications, 2nd ed. London: Springer-Verlag.
  • Brouwer, A. E., Haemers, W. H. (2011). Spectra of Graphs. New York: Springer-Verlag.
  • Bubboloni, D., Iranmanesh, M. A., Shaker, S. M. (2017). On some graphs associated with the finite alternating groups. Commun. Algebra. 45(12): 5355–5373.
  • Cameron, P. J. (1999). Permutation Groups, London Mathematical Society Student Texts. Vol. 45. Cambridge, UK: Cambridge University Press.
  • Cameron, P. J. (2010). The power graph of a finite group II. J. Group Theory 13(6): 779–783.
  • Cameron, P. J. Graphs defined on groups, Internat. J. Group Theory, in press. https://dx.doi.org/10.22108/ijgt.2021.127679.1681
  • Cameron, P. J., Ghosh, S. (2011). The power graph of a finite group. Discret. Math. 311(13): 1220–1222.
  • Cameron, P. J., Guerra, H., Jurina, Š. (2019). The power graph of a torsion-free group. J. Algebr. Comb. 49(1): 83–98.
  • Cameron, P. J., Jafari, S. H. (2020). On the connectivity and independence number of power graphs of groups. Graphs Comb. 36(3): 895–904.
  • Cameron, P. J., Manna, P., Mehatari, R. (2021). Forbidden subgraphs of power graphs. Electron. J. Combinat. 28(3).
  • Cameron, P. J., Maslova, N. V. Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph. Available at: https://arxiv.org/abs/2012.04812.
  • Chakrabarty, I., Ghosh, S., Sen, M. K. (2009). Undirected power graphs of semigroups. Semigroup Forum. 78(3): 410–426.
  • Chattopadhyay, S., Panigrahi, P. (2014). Connectivity and planarity of power graphs of finite cyclic, dihedral and dicyclic groups. Alge. Discrete Math. 18(1): 42–49.
  • Chattopadhyay, S., Panigrahi, P. (2015). On laplacian spectrum of power graphs of finite cyclic and dihedral groups. Linear Multilinear Algebra 63(7): 1345–1355.
  • Chattopadhyay, S., Panigrahi, P. (2015). Some relations between power graphs and Cayley graphs. J. Egyptian Math. Soc. 23(3): 457–462.
  • Chattopadhyay, S., Panigrahi, P. (2017). On sum of powers of the laplacian eigenvalues of power graphs of certain finite groups, Electron. Notes Discret. Math. 63: 137–143.
  • Chattopadhyay, S., Panigrahi, P., Atik, F. (2018). Spectral radius of power graphs on certain finite groups. Indag. Math. 29(2): 730–737.
  • Chattopadhyay, S., Patra, K. L., Sahoo, B. K. (2019). Vertex connectivity of the power graph of a finite cyclic group. Discret. Appl. Math. 266: 259–271.
  • Chattopadhyay, S., Patra, K., Sahoo, B. (2020). Erratum: Vertex connectivity of the power graph of a finite cyclic group II. J. Algebra Appl. 19(02): 2050040.
  • Chattopadhyay, S., Patra, K. L., Sahoo, B. K. (2020). Minimal cut-sets in the power graphs of certain finite non-cyclic groups. Comm. Algebra. 49(3): 1–17.
  • Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R. (2006). The strong perfect graph theorem. Ann. Math. 164(1): 51–229.
  • Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A, Wilson, R. A. (1985). An ATLAS of Finite Groups. Oxford: Oxford Univ. Press.
  • Curtin, B., Pourgholi, G. R. (2014). Edge-maximality of power graphs of finite cyclic groups. J. Algebr. Comb. 40(2): 313–330.
  • Curtin, B., Pourgholi, G. R., Yousefi-Azari, H. (2015). On the punctured power graph of a finite group. Australas. J. Combinatorics 62(1): 1–7.
  • Cvetkovic, D., Simic, S., Rowlinson, P. (2009). An Introduction to the Theory of Graph Spectra. Cambridge: Cambridge University Press.
  • Cvetkovic, D. M., Rowlinson, P., (2010). and S. Simic, An Introduction to the Theory of Graph Spectra. Cambridge, UK: Cambridge University Press.
  • Dilworth, R. P. (1950). A decomposition theorem for partially ordered sets. Ann. Math 51(1): 161–166.
  • Doostabadi, A., Erfanian, A., Farrokhi D.G, M. (2014). On power graphs of finite groups with forbidden induced subgraphs. Indag. Math 25(3): 525–533.
  • Doostabadi, A., Farrokhi, D., Ghouchan, M. (2015). On the connectivity of proper power graphs of finite groups. Comm. Algebra 43(10): 4305–4319.
  • Dummit, D. S., Foote, R. M. (2004). Abstract Algebra, 3rd ed. Hoboken, NJ: Wiley.
  • Feng, M., Ma, X., Wang, K. (2015). The structure and metric dimension of the power graph of a finite group. Eur. J. Combinat. 43: 82–97.
  • Feng, M., Ma, X, Wang, K. (2016). The full automorphism group of the power (di)graph of a finite group. Eur. J. Combinat. 52: 197–206.
  • Fiedler, M. (1973). Algebraic connectivity of graphs. Czech. Math. J. 23(2): 298–305.
  • Gallian, J. A. (1999). Contemporary Abstract Algebra, 4th ed. India: Narosa Publishing House.
  • The GAP group, GAP – groups, algorithms, and programming. Available at: http://www.gap-system.org/
  • Hall, M. Jr., (1959). The Theory of Groups. New York: Macmillan.
  • Hell, P., Nešetřil, J. (2004). Graphs and Homomorphisms, Oxford Lecture Series in Mathematics and Its Applications. Vol. 28. Oxford: Oxford University Press.
  • Hammer, P. L., Foldes, S. (1977). Split graphs, in: Proceedings of the 8th South-Eastern Conference on Combinatorics. Graph Theory and Computing, Congressus Numeratum, Vol. XIX, 311–315.
  • Hamzeh, A, Ashrafi, A. R. (2017). Spectrum and l-spectrum of the power graph and its main supergraph for certain finite groups. spectrum and FILOMAT 31(16): 5323–5334.
  • Hungerford, T. W. (1974). Algebra Graduate Texts in Mathematics. Vol. 73, New York: Springer.
  • Jafari, S. H. (2015). Some results in a new power graphs in finite groups. Util. Math 103: 181–187.
  • James, G., Liebeck, M. (2001). Representations and Characters of Groups, 2nd ed. Cambridge: Cambridge Univ. Press.
  • Kelarev, A. V, Quinn, S. J. (2000). A combinatorial property and power graphs of groups. Contrib. General Algebra 12(58): 3–6.
  • Kelarev, A. V., Quinn, S. J. (2002). Directed graphs and combinatorial properties of semigroups. J. Algebra 251(1): 16–26.
  • Kelarev, A. V., Quinn, S. J., Smolikova, R. (2001). Power graphs and semigroups of matrices. Bull. Austral. Math. Soc. 63(2): 341–344.
  • Kirkland, S. J., Molitierno, J. J., Neumann, M., Shader, B. L. (2002). On graphs with equal algebraic and vertex connectivity. Linear Algebra Appl. 341(1–3): 45–56.
  • Li, X. (2012). Graph Energy. New York: Springer-Verlag
  • Lovász, L. (1972). Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2(3): 253–267.
  • Ma, X., Feng, M. (2015). On the chromatic number of the power graph of a finite group. Indag. Math. 26(4): 626–633.
  • Ma, X., Feng, M., Wang, K. (2016). The rainbow connection number of the power graph of a finite group. Graph. Combinator. 32(4): 1495–1504.
  • Ma, X., Fu, R., Lu, X. (2018). On the independence number of the power graph of a finite group. Indag. Math. 29(2): 794–806.
  • Ma, X., Walls, G. L., Wang, K. (2019). Power graphs of (non)orientable genus two. Comm. Algebra 47(1): 276–288.
  • McKemmie, E. (2014). Power graphs of finite groups. BA Project Supervised by Peter M. Neumann at Oxford University.
  • Mehranian, Z., Gholami, A., Ashrafi, A. R. (2016). A note on the power graph of a finite group. Int. J. Group Theory 5(1): 1–10.
  • Mehranian, Z., Gholami, A, Ashrafi, A. R. (2017). The spectra of power graphs of certain finite groups. Linear Multilinear Algebra 65(5): 1003–1010.
  • Mirzargar, M., Ashrafi, A. R., Nadjafi-Arani, M. J. (2012). On the power graph of a finite group. FILOMAT. 26(6): 1201–1208.
  • Moghaddamfar, A. R., Rahbariyan, S, Shi, W. J. (2014). Certain properties of the power graph associated with a finite group. J. Algebra Appl. 13(07): 1450040.
  • Mohar, B. (1991). The Laplacian spectrum of graphs. In: ed. Alavi, Y., Chartrand, G., Oellermann, O. R., Schwenk, A. J. Graph Theory, Combinatorics, and Applications Vol. 2, 871–898. New York: Wiley.
  • Mukherjee, H. (2019). Hamiltonian cycles of power graph of Abelian groups. Afr. Mat. 30(7/8): 1025–1040.
  • Panda, R. P. (2019). Laplacian spectra of power graphs of certain finite groups. Graph. Combinator 35(5): 1209–1223.
  • Panda, R. P. (2020). A combinatorial characterization of finite groups of prime exponent. Indag. Math. 31(1): 1–6.
  • Panda, R. P., Krishna K. V. (2018). On connectedness of power graphs of finite groups. J. Algebra Appl. 17(10): 1850184.
  • Panda, R. P., Krishna, K. V. (2018). On the minimum degree, edge-connectivity and connectivity of power graphs of finite groups. Comm. Algebra 46(7): 3182–3197.
  • Panda, R. P., Patra, K. L., Sahoo, B. K. (2021). On the minimum degree of the power graph of a finite cyclic group. J. Algebra Appl. 20(03): 2150044.
  • Pourghobadi, K, Jafari, S. H. (2018). The diameter of proper power graphs of symmetric groups. J. Algebra Appl. 17(12): 1850234.
  • Pourgholi, G. R., Yousefi-Azari, H., Ashrafi, A. R. (2015). The undirected power graph of a finite group. Bull. Malays. Math. Sci. Soc. 38(4): 1517–1525.
  • Raj, A., Singh, S. N. (2018). The Laplacian spectrum of power graphs of some finite abelian p-groups. Available at: https://arxiv.org/abs/1802.0805v2.
  • Rose, J. S. (1994). A Course on Group Theory. New York: Dover Publications.
  • Sehgal, A., and Singh, S. N. (2019). The degree of a vertex in the power graph of a finite abelian group. Available at: https://arxiv.org/abs/1901.08187v2.
  • Shitov, Y. (2017). Coloring the power graph of a semigroup. Graph. Combinator 33(2): 485–487.
  • Tamizh Chelvam, T., Sattanathan, M. (2013). Power graph of finite abelian groups. Algebra Discret. Math 16: 33–41.
  • Williams, J. S. (1981). Prime graph components of finite groups. J. Algebra 69(2): 487–513.
  • Zahirović, S. (2019). The power graph of a torsion-free group of nilpotency class 2. Available at: https://arxiv.org/abs/1911.00555v2
  • Zahirović, S. The power graph of a torsion-free group determines the directed power graph. Available at: https://arxiv.org/abs/2006.01984