A note on the power graph of a finite group

Document Type: Research Paper

Authors

1 Univ Qom

2 University of Kashan

Abstract

‎Suppose $\Gamma$ is a graph with $V(\Gamma) = \{ 1‎, ‎2,\dots‎, ‎p\}$‎ ‎and $ \mathcal{F} = \{\Gamma_1,\dots‎, ‎\Gamma_p\} $ is a family of‎ ‎graphs such that $n_j = |V(\Gamma_j)|$‎, ‎$1 \leq j \leq p$‎. ‎Define‎ ‎$\Lambda = \Gamma[\Gamma_1,\dots‎, ‎\Gamma_p]$ to be a graph with‎ ‎vertex set $ V(\Lambda)=\bigcup_{j=1}^pV(\Gamma_j)$ and edge set‎ ‎$E(\Lambda)=\big(\bigcup_{j=1}^pE(\Gamma_j)\big)\cup\big(\bigcup_{ij\in‎ ‎E(\Gamma)}\{uv;u\in V(\Gamma_i),v\in V(\Gamma_j)\}\big) $‎. ‎The‎ ‎graph $ \Lambda$ is called the $\Gamma$-join of $ \mathcal{F}$‎. ‎The power graph $\mathcal{P}(G)$ of a group $G$ is the graph‎ ‎which has the group elements as vertex set and two elements are‎ ‎adjacent if one is a power of the other‎. ‎The aim of this paper is‎ ‎to prove that $\mathcal{P}(\mathbb{Z}_{n}) = K_{\phi(n)+1}‎ + ‎\Delta_n[K_{\phi(d_1)}‎, ‎K_{\phi(d_2)},\dots‎, ‎K_{\phi(d_{p})}]$‎, ‎where $\Delta_n$ is a graph with vertex and edge sets‎ ‎$V(\Delta_n)=\{d_i \ | \ 1,n\not = d_i | n‎, ‎1\leq i\leq p\}$ and‎ ‎$ E(\Delta_n)=\{ d_id_j \ | \ d_i|d_j‎, ‎1\leq i<j\leq p\}$‎, ‎respectively‎. ‎As a consequence it is proved that‎ ‎$Aut(\mathcal{P}(\mathbb{Z}_{n}))\cong‎ ‎S_{\phi(n)+1}\times\prod_{1,n\not=d|n}S_{\phi(d)}.$ This proves a‎ ‎recent conjecture by Doostabadi et al‎. ‎[A‎. ‎Doostabadi‎, ‎A‎. ‎Erfanian and A‎. ‎Jafarzadeh‎, ‎Some results on the power graph of groups, ‎The Extended Abstracts of the 44th Annual Iranian Mathematics Conference‎, ‎27-30 August 2013‎, ‎Ferdowsi University of Mashhad‎, ‎Iran]‎. ‎Finally‎, ‎we‎ ‎apply our results to obtain complete descriptions of the power‎ ‎graphs of some finite groups‎.

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P‎. ‎J‎. ‎Cameron and S‎. ‎Ghosh (2011). ‎The power‎ ‎graph of a finite group. Discrete Math.. 311, 1220-1222
P‎. ‎J‎. ‎Cameron (2010). ‎The power graph of a finite group. J‎. ‎Group Theory. 13, 779-783
D‎. ‎M‎. ‎Cardoso‎, ‎M‎. ‎A‎. ‎de Freitas‎, ‎E‎. ‎A‎. ‎Martins and M‎. ‎Robbiano (2013). ‎Spectra of graphs obtained by a generalization of the join graph operation. Discrete Math.. 313, 733-741
I‎. ‎Chakrabarty‎, ‎S‎. ‎Ghosh and M‎. ‎K‎. ‎Sen (2009). ‎Undirected‎ ‎power graphs of semigroups. Semigroup Forum. 78, 410-426
A‎. ‎Doostabadi‎, ‎A‎. ‎Erfanian and A‎. ‎Jafarzadeh (2013). Some results on the power graph of groups. ‎The Extended Abstracts of the 44th Annual Iranian Mathematics Conference‎, ‎Ferdowsi University of Mashhad‎, ‎Iran. , 27-30
A‎. ‎V‎. ‎Kelarev and S‎. ‎J‎. ‎Quinn (2000). A combinatorial property and power graphs of groups. ‎Contributions to General Algebra‎, ‎‎Heyn‎, ‎Klagenfurt. 12, 229-235
A‎. ‎Kelarev‎, ‎J‎. ‎Ryan and J‎. ‎Yearwood (2009). ‎Cayley graphs as classifiers for data mining‎: ‎the influence of asymmetries. Discrete Math.. 309, 5360-5369
A‎. ‎V‎. ‎Kelarev (2004). ‎Labelled Cayley graphs and minimal automata. Australas‎. ‎J‎. ‎Combin.. 30, 95-101
A‎. ‎V‎. ‎Kelarev (2003). Graph Algebras and Automata. ‎Marcel Dekker‎, ‎New York. 257
A‎. ‎V‎. ‎Kelarev‎, ‎S‎. ‎J‎. ‎Quinn and R‎. ‎Smolikova (2001). ‎Power graphs‎ ‎and semigroups of matrices. Bull‎. ‎Austral‎. ‎Math‎. ‎Soc.. 93, 341-344
A‎. ‎V‎. ‎Kelarev and S‎. ‎J‎. ‎Quinn (2002). ‎Directed graphs and combinatorial properties of semigroups. J‎. ‎Algebra. 251 (1), 16-26
A‎. ‎V‎. ‎Kelarev and S‎. ‎J‎. ‎Quinn (2004). ‎A combinatorial property and power graphs of semigroups. Comment‎. ‎Math‎. ‎Univ‎. ‎Carolin.. 45, 1-7
J‎. ‎Abawajy‎, ‎A‎. ‎V‎. ‎Kelarev and M‎. ‎Chowdhury (2013). ‎Power graphs‎: ‎a survey. Electron‎. ‎J‎. ‎Graph Theory Appl‎. ‎(EJGTA). 1 (2), 125-147
M‎. ‎Mirzargar‎, ‎A‎. ‎R‎. ‎Ashrafi and M‎. ‎J‎. ‎Nadjafi-Arani (2012). ‎On the power graph of a finite group. Filomat. 26, 1201-1208
A‎. ‎R‎. ‎Moghaddamfar‎, ‎S‎. ‎Rahbariyan and W‎. ‎J‎. ‎Shi (2014). ‎Certain properties of the power graph associated with a finite group. J‎. ‎Algebra Appl.. 13 (7), 0
A‎. ‎R‎. ‎Moghaddamfar‎, ‎S‎. ‎Rahbariyan‎, ‎S‎. ‎Navid Salehy and S‎. ‎Nima Salehy ‎The number of spanning trees of power graphs associated with specific groups and some applications. ‎to appear in, Ars Combinatoria.
G‎. ‎R‎. ‎Pourgholi‎, ‎H‎. ‎Yousefi-Azari and A‎. ‎R‎. ‎Ashrafi ‎The undirected power graph of a finite group. ‎to appear in, Bull‎. ‎Malaysian Math‎. ‎Sci‎. ‎Soc..
J‎. ‎S‎. ‎Rose (1994). A Course on Group Theory. ‎Dover Publications‎, ‎Inc.‎, ‎New York.