A survey on enhanced power graphs of finite groups
Abstract
We survey known results on enhanced power graphs of finite groups. Open problems, questions and suggestions for future work are also included.
Keywords
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PDFDOI: http://dx.doi.org/10.5614/ejgta.2022.10.1.6
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), #P3.16.
J. Abawajy, A. Kelarev, and M. Chowdhury, Power graphs: A survey, Electron. J. Graph Theory Appl. 1 (2013), 125–147.
A. Abdollahi and A.M. Hassanabadi, Noncyclic graph of a group, Commun. Algebra 35 (2007), 2057–2081.
A. Abdollahi and A.M. Hassanabadi, Noncyclic graph associated with a group, J. Algebra Appl. 8 (2009), 243–257.
M. Afkhami, A. Jafarzadeh, K. Khashyarmanesh, and S. Mohammadikhah, On cyclic graphs of finite semigroups, J. Algebra Appl. 13 (2014), 1450035, 11 pp.
S. Akhter and R. Farooq, Metric dimension of fullerene graphs, Electron. J. Graph Theory Appl. 7 (2019), 91–103.
R. Alfarisi, Y. Lin, J. Ryan, D. Dafik, and I.H. Agustin, A note on multiset dimension and local multiset dimension of graphs, Statistics, Optimization & Information Computing 8 (2020), 890–901.
R. Alfarisi, J. Ryan, M.K. Siddiqui, Dafik, and I.H. Agustin, Vertex irregular reflexive labeling of disjoint union of gear and book graphs, Asian-European Journal of Mathematics 14 (2021), 2150078.
M. Baca, E.T. Baskoro, Y. Lin, O. Phanalasy, J. Ryan, and A. Semanicová-Fenovciková, A survey of face-antimagic evaluations of graphs, Australasian J. Combinatorics 69 (2017), 382–393.
M. Baca, M. Irfan, A. Javed, and A. Semanicova-Fenovcikova, On total edge product cordial labeling of fullerenes, Electron. J. Graph Theory Appl. 6 (2018), 238–249.
H.H. Bashir and K. Ahmadidelir, Some structural graph properties of the non-commuting graph of a class of finite Moufang loops, Electron. J. Graph Theory Appl. 8 (2020), 319–337.
E.T. Baskoro and D.I.D. Primaskun, Improved algorithm for the locating-chromatic number of trees, Theoret. Comp. Sci. 856 (2021), 165–168.
C. Bazgan, L. Brankovic, K. Casel, H. Fernau, K. Jansen, K.M. Klein, M. Lampis, M. Liedloff, J. Monnot, and V.T. Paschos, Algorithmic aspects of upper domination: a parameterised perspective, Int. Conf. Algorithmic Applications in Management, AAM 2016, 113–124.
S. Bera, On the intersection power graph of a finite group, Electron. J. Graph Theory Appl. 6 (2018), 178–189.
S. Bera and A.K. Bhuniya, On enhanced power graphs of finite groups, J. Algebra Appl. 17 (2018), 1850146, 8 pp.
S. Bera and H.K. Dey, On connectivity, domination number and spectral radius of the proper enhanced power graphs of finite nilpotent groups, ArXiv (2021), https://arxiv.org/abs/2108.05175v1.
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.
Z.N. Berberler and M.E. Berberler, Independent strong domination in complementary prisms, Electron. J. Graph Theory Appl. 8 (2020), 1–8.
N.H. Bong and Y. Lin, Some properties of the multiset dimension of graphs, Electron. J. Graph Theory Appl. 9 (2021), 215–221.
I. Bošnjak, R. Madarász, and S. Zahirović, Some new results concerning power graphs and enhanced power graphs of groups, ArXiv (2020), https://arxiv.org/abs/2012.02851v1.
R. Brauer and K.A. Fowler, On groups of even order, Ann. Math. 62 (1955), 565–583.
P.J. Cameron, The power graph of a finite group, II, J. Group Theory 13 (2010), 779–783.
P.J. Cameron, Graphs defined on groups, Int. J. Group Theory, in press, doi.org/110.22108/ijgt.2021.127679.1681.
P.J. Cameron and S. Ghosh, The power graph of a finite group, Discrete Math. 311 (2011), 1220–1222.
P.J. Cameron and B. Kuzma, Between the enhanced power graph and the commuting graph, ArXiv (2021), https://arxiv.org/abs/2012.03789v1.
A. Cayley, Desiderata and suggestions: No. 2. The theory of groups: graphical representation, American J. Math. 1 (1878), 174–176.
I. Chakrabarty, S. Ghosh, and M.K. Sen, Undirected power graphs of semigroups, Semigroup Forum 78 (2009), 410–426.
G. Chartrand, D. Erwin, M.A. Henning, P.J. Slater, and P. Zhang, The locating-chromatic number of a graph, Bull. Inst. Combin. Appl. 36 (2002), 89–101.
N. Chikh and M. Mihoubi, Note on chromatic polynomials of the threshold graphs, Electron. J. Graph Theory Appl. 7 (2019), 217–224.
D.G. Costanzo and M.L. Lewis, The cyclic graph of a 2-Frobenius group, ArXiv (2021), https://arxiv.org/abs/2103.15574v1.
D.G. Costanzo, M.L. Lewis, S. Schmidt, E. Tsegaye, and G. Udell, The cyclic graph (deleted enhanced power graph) of a direct product, Involve 14 (2021), 167–179.
D.G. Costanzo, M.L. Lewis, S. Schmidt, E. Tsegaye, and G. Udell, The cyclic graph of a Z-group, Bull. Aust. Math. Soc., 104 (2021), 295–301.
S. Dalal and J. Kumar, Chromatic number of the cyclic graph of infinite semigroup, Graphs Combin. 36 (2020), 109–113.
S. Dalal and J. Kumar, On enhanced power graphs of certain groups, Discrete Math. Algorithms Appl. 13 (2021), 2050099, 18 pp.
S. Dalal, J. Kumar, and S. Singh, On the enhanced power graph of a semigroup, ArXiv (2021), https://arxiv.org/abs/2107.11793v1.
S. Dalal, J. Kumar, and S. Singh, The cyclic graph of a semigroup, ArXiv (2021), https://arxiv.org/abs/2107.11021v1.
A. Das, Connected domination value in graphs, Electron. J. Graph Theory Appl. 9 (2021), 113–123.
L.A. Dupont, D.G. Mendoza, and M. Rodríguez, The rainbow connection number of enhanced power graph, ArXiv (2017), https://arxiv.org/abs/1708.07598v1.
L.A. Dupont, D.G. Mendoza, and M. Rodríguez, The enhanced quotient graph of the quotient of a finite group, ArXiv (2017), https://arxiv.org/abs/1707.01127v1.
L.A. Dupont, R. López, and M. Rodriguez, The rainbow k-connectivity of the non-commutative graph of a finite group, Electron. J. Graph Theory Appl. 8 (2020), 93–111.
H.T. Faal, Clique roots of K4-free chordal graphs, Electron. J. Graph Theory Appl. 7 (2019), 105–111.
D.A. Fahrudin and S.W. Saputro, The geodetic domination number of comb product graphs, Electron. J. Graph Theory Appl. 8 (2020), 373–381.
M. Feng, X. Ma, and K. Wang, The structure and metric dimension of the power graph of a finite group, Eur. J. Combin. 43 (2015), 82–97.
F. Harary and R.A. Melter, On the metric dimension of a graph, Ars Combinatoria 2 (1976), 191–195.
S.M. Hosamani, S. Shirkol, P.B. Jinagouda, and M. Krzywkowski, Degree equitable restrained double domination in graphs, Electron. J. Graph Theory Appl. 9 (2021), 105–111.
D. Indriati, W. Widodo, I.E. Wijayanti, K.A. Sugeng and I. Rosyida, Totally irregular total labeling of some caterpillar graphs, Electron. J. Graph Theory Appl. 8 (2020), 247–254.
A.V. Kelarev, Ring Constructions and Applications, World Scientific, 2002.
A.V. Kelarev, Graph Algebras and Automata, Marcel Dekker, New York, 2003.
A.V. Kelarev and S.J. Quinn, A combinatorial property and power graphs of groups, Proc. of the Vienna conference 1999, Contrib. General Algebra 12 (2000), 229–235.
A.V. Kelarev and S.J. Quinn, Directed graphs and combinatorial properties of semigroups, J. Algebra 251 (2002), 16–26.
A.V. Kelarev and S.J. Quinn, A combinatorial property and power graphs of semigroups, Comment. Math. Univ. Carolinae 45 (2004), 1–7.
A.V. Kelarev, S.J. Quinn, and R. Smolikova, Power graphs and semigroups of matrices, Bull. Austral. Math. Soc. 63 (2001), 341–344
A.V. Kelarev, J. Ryan, and J. Yearwood, Cayley graphs as classifiers for data mining: The influence of asymmetries, Discrete Math. 309 (2009), 5360–5369.
S.M.S. Khasraw, I.D. Ali, and R.R. Haji, On the non-commuting graph of dihedral group, Electron. J. Graph Theory Appl. 8 (2020), 233–239.
A. Kumar, L. Selvaganesh, P.J. Cameron, and T.T. Chelvam, Recent developments on the power graph of finite groups – a survey, AKCE Int. J. Graphs Comb. 18 (2021), 65–94.
D. Kuziak, I.G. Yero and J.A.Rodríguez-Velázquez, On the strong metric dimension of the strong products of graphs, Open Math. 13 (2015), 64–74.
S. Mitra and S. Bhoumik, Graceful labeling of triangular extension of complete bipartite graph, Electron. J. Graph Theory Appl. 7 (2019), 11–30.
D.A. Mojdeh, S.R. Musawi, and E. Nazari, On the distance domination number of bipartite graphs, Electron. J. Graph Theory Appl. 8 (2020), 353–364.
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.
X. Ma, Y. Lv, and Y. She, Forbidden subgraphs in enhanced power graphs of finite groups, ArXiv (2021), https://arxiv.org/abs/2104.04754v3.
X. Ma, H. Wei, and G. Zhong, The cyclic graph of a finite group, Algebra (2013), 107265, 7 pp.
X. Ma and L. Zhai, Strong metric dimensions for power graphs of finite groups, Commun. Algebra, 49 (2021), 4577–4587.
A. Mahmoudifar and A. Babai, On the structure of finite groups with dominatable enhanced power graph, J. Algebra Appl. 21 (2022), 2250176, 8 pp.
T.K. Maryati, A.N.M. Salman, E.T. Baskoro, J. Ryan, and M. Miller, On H-supermagic labelings for certain shackles and amalgamations of a connected graph, Utilitas Mathematica 83 (2010), 333–342.
B.H. Neumann, A problem of Paul Erdös on groups, J. Aust. Math. Soc. (Ser. A) 21 (1976), 467–472.
A.A.G. Ngurah and R. Simanjuntak, On distance labelings of 2-regular graphs, Electronic J. Graph Theory & Appl. (EJGTA) 9 (2021), 25–37.
K. O’Bryant, D. Patrick, L. Smithline, and E. Wepsic, Some facts about cycles and tidy groups, Tech. Rep. MS-TR 92–04, Rose–Hulman Institute of Technology, Terre Haute, USA, 1992.
L. Ouldrabah, M. Blidia, and A. Bouchou, Roman domination in oriented trees, Electronic J. Graph Theory & Appl. (EJGTA) 9 (2021), 95–103.
P.R. Panda, S. Dalal, and J. Kumar, On the enhanced power graph of a finite group, Comm. Algebra 49 (2021), 1697–1716.
D. Patrick and E. Wepsic, Cyclicizers, centralizers and normalizers, Tech. Rep. MS-TR 91–05, Rose-Hulman Institute of Technology, Terre Haute, USA, 1991.
H.M. Radiapradana, S.W. Saputro, E. Suwastika, O. Neswan, and A. Semanicova-Fenovcikova, On topological integer additive set-labeling of star graphs, Electronic J. Graph Theory & Appl. (EJGTA) 6 (2018), 341–346.
A. Seb and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2004), 383–393.
R. Simanjuntak, P. Siagian, and T. Vetrik, The multiset dimension of graphs, ArXiv (2017), https://arxiv.org/abs/1711.00225.
P.J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549–559.
D. Tanna, J. Ryan, A. Semanicová-Fenovciková, and M. Baca, Vertex irregular reflexive labeling of prisms and wheels, AKCE Int. J. Graphs & Combinatorics 17 (2020), 51–59.
The GAP Team, GAP — Groups, Algorithms, Programming - a System for Computational Discrete Algebra, 2021, https://www.gap-system.org/, cited 24 August 2021.
The Magma Team, Magma Computational Algebra System, 2021, http://magma.maths.usyd.edu.au/magma/, cited 24 August 2021.
The SageMath Team, SageMath — Open-Source Mathematical Software System, 2021, https://www.sagemath.org, cited 24 August 2021.
E. Vatandoost and M. Khalili, Domination number of the non-commuting graph of finite groups, Electronic J. Graph Theory & Appl. (EJGTA) 6 (2018), 228–237.
S. Zahirović, I. Bošnjak, and R. Madarász, A study of enhanced power graphs of finite groups, J. Algebra Appl. 19 (2020), 2050062, 20 pp.
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