### A SURVEY ON THE AUTOMORPHISM GROUPS OF THE COMMUTING GRAPHS AND POWER GRAPHS

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#### Abstract

Let G be a nite group. The power graph P(G) of a group G is the graph

whose vertex set is the group elements and two elements are adjacent if one is a power of the other. The commuting graph \Delta(G) of a group G, is the graph whose vertices are the group elements, two of them joined if they commute. When the vertex set is G-Z(G), this graph is denoted by \Gamma(G). Since the results based on the automorphism group of these kinds of graphs are so sporadic, in this paper, we give a survey of all results on the automorphism group of power graphs and commuting graphs obtained in the literature.

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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. Comb. 24 (2017), 3-16.

J. Abawajy, A. Kelarev and M. Chowdhury: Power graphs: a survey. Electron. J. Graph Theory Appl. (EJGTA) 1 (2013), 125-147.

A. Abdollahi, S. Akbari and H. R. Maimani :Non-commuting graph of a group. J. Algebra 298 (2006), 468-492.

A. Abdollahi and H. Shahverdi :Non-commuting graphs of nilpotent groups. Commun. Algebra 42 (2014), 3944-3949.

A. R. Ashrafi, A. Gholami and Z. Mehranian :Automorphism group of certain power graphs of nite groups. Electron. J. Graph Theory Appl. (EJGTA) 5 (2017),70-82.

J. Bosak :The graphs of semigroups, Theory of Graphs and Application. Academic Press, New York, 1964.

F. Budden: Cayley graphs for some well-known groups. Math. Gaz. 69 (1985),271-278.

P. J. Cameron and S. Ghosh: The power graph of a nite group. Discrete Math. 311 (2011), 1220-1222.

P. J. Cameron: The power graph of a nite group II. J. Group Theory, 13 (2010), 779-783.

P. J. Cameron: Permutation Groups. Cambridge Univ. Press, Cambridge, 1999.

I. Chakrabarty, S. Ghosh and M. K. Sen:Undirected power graphs of semi-groups. Semigroup Forum 78 (2009), 410-426.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wil-son: Atlas of nite simple groups, Maximal subgroups and ordinary characters for simple groups. Oxford University Press, Eynsham, 1985.

B. Zelinka:Intersection graphs of nite Abelian groups. Czech. Math. J. 25 (1975), 171-174.

M. R. Darafsheh: Groups with the same non-commuting graph. Discrete Applied Math. 157 (2009), 833-837.

M. R. Darafsheh and N. S. Poursalavati: On the existence of the orthogonal basis of the symmetry classes of tensors associated with certain groups. Sut J. Math. 37 (2001), 1-17.

A. Doostabadi, A. Erfanian and A. Jafarzadeh: Some results on the power graph of groups.In: The Extended Abstracts of the 44th Annual Iranian Mathematics Conference, Ferdowsi University of Mashhad, Iran, 2013, pp. 27-30.

M. Feng, X. Ma and K. Wang:The full automorphism group of the power (di)graph of a nite group. European J. Combin. 52 (2016), 197-206.

R. Frucht: On the groups of repeated graphs. Bull. Amer. Math. Soc. 55 (1949), 418-420.

A. Hamzeh and A. R. Ashrafi:Automorphism groups of supergraphs of the power graph of a nite group. Eur. J. Comb. 60 (2017), 82-88.

G. James and M. Liebeck: Representations and Characters of Groups. 2nd ed., Cambridge University Press, New York, 2001.

J. A. Gallian: Contemporary Abstract Algebra. Narosa Publishing House,London, 1999.

A. V. Kelarev and S. J. Quinn: Directed graph and combinatorial properties of semigroups. J. Algebra 251 (2002), 16-26.

Z. Mehranian, A. Gholami and A. R. Ashrafi: A note on the power graph of a nite group. Int. J. Group Theory 5 (2016), 1-10.

M. Mirzargar and A. R. Ashrafi: Some distance-based topological indices of the non-commuting graph. Hacet. J. Math. Stat. 41 (2012), 515-526.

M. Mirzargar, A. R. Ashrafi and M. J. Nadjafi-Arani: On the power graph of a nite group. Filomat 26 (2012), 1201{1208.

M. Mirzargar, P. P. Pach and A. R. Ashrafi: The automorphism graph of commuting graph of a nite group. Bull. Korean Math. Soc. 51 (2014), 1145-1153.

M. Mirzargar, P. P. Pach and A. R. Ashrafi: Remarks On Commuting Graph of a Finite Group. Electron. Notes Discrete Math. 45 (2014), 103-106.

A. R. Moghaaddamfar: About noncommuting graphs, Siberian Math. J. 47 (2006), 911-914.

B. H. Neumann, A problem of Paul Erdos on groups. J. Aust. Math. Soc. Ser. 21 (1976), 467-472.

G. R. Pourgholi and H. Yousefi-Azari:On the 2-connected power graphs of nite groups. Australiasian J. Combinatorics 62 (2015), 1-7.

D. M. Rocke: p-groups with abelian centralizers. Proc. London Math. Soc. 30 (1975), 55-75.

H. E. Rose: A Course on Finite Groups. Cambridge University press, Cam-bridge, 1978.

D. Witte, G. Letzter and J. A. Gallian: On Hamiltonian circuits in Carte-sian products of Cayley digraphs. Discrete Math. 43 (1983), 297-307.

M. Mirzargar, P. P. Pach and A. R. Ashrafi: Remarks On Commuting Graph of a Finite Group. Electron. Notes Discrete Math. 45 (2014), 103–106.

A. R. Moghaaddamfar: About noncommuting graphs, Siberian Math. J. 47 (2006), 911–914.

B. H. Neumann, A problem of Paul Erdos on groups. J. Aust. Math. Soc. Ser. 21 (1976), 467–472.

G. R. Pourgholi and H. Yousefi-Azari:On the 2-connected power graphs of finite groups. Australiasian J. Combinatorics 62 (2015), 1–7.

D. M. Rocke: p-groups with abelian centralizers. Proc. London Math. Soc. 30 (1975), 55–75.

H. E. Rose: A Course on Finite Groups. Cambridge University press, Cambridge, 1978.

D. Witte, G. Letzter and J. A. Gallian: On Hamiltonian circuits in Cartesian products of Cayley digraphs. Discrete Math. 43 (1983), 297–307.

DOI: https://doi.org/10.22190/FUMI1904729M

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