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Year 2018, Volume: 47 Issue: 1, 93 - 99, 01.02.2018

Abstract

References

  • A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006), 468-492.
  • A. Abdollahi, Engel graph associated with a group, J. Algebra, 318 (2007), 680-691.
  • A. Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group, Comm. in Algebra, 35 (2007), 2057-2081.
  • R. Barzegar and A. Erfanian, Nilpotency and solubility of groups relative to an automorphism, Caspian Journal of Mathematical Sciences, 4(2) (2015), 271-283.
  • A. Erfanian, M. Farrokhi D.G. and B. Tolue, Non-normal graphs of finite groups, J. Algebra Appl., 12 (2013).
  • J. A. Bondy and J. S. R. Murty, Graph Theory with Applications, Elsevier, (1977).
  • P. J. Cameron and S. Ghosh, The power graph of a fnite group, Discrete Math, 311 (2011), 1220-1222.
  • I. Fabrici and T. Madaras, The structure of 1-planar graphs, Discrete Mathematics, 307 (2007), 854 - 865.
  • A. Iranmanesh and A. Jafarzadeh, On the commuting graph associated with the symmetric and alternating groups, J. Algebra Appl., 7 (2008), 129-146.
  • D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag: New York-Heidelberg Berlin (1982).
  • The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.6.4, 2013 (http://www.gap-system.org/).
  • J. S. Williams, Prime graph components of finite groups, J. Algebra, 69 (2) (1981), 487-513.

A graph associated to a fixed automorphism of a finite group

Year 2018, Volume: 47 Issue: 1, 93 - 99, 01.02.2018

Abstract

Let $G$ be a finite group and $Aut(G)$ be the group of automorphisms of $G$. We associate a graph to a group $G$ and fixed automorphism $\alpha$ of $G$ denoted by $\Gamma_G^\alpha$. The vertex set of $\Gamma_G^\alpha$ is $G\backslash Z^\alpha(G)$ and two vertices $x,g\in G\backslash Z^\alpha(G)$ are adjacent if $[g,x]_\alpha\neq 1$ or $[x,g]_\alpha\neq 1$, where $[g,x]_\alpha=g^{-1}x^{-1}gx^\alpha$ and $Z^\alpha(G)=\{ x\in G\,|\, [g,x]_\alpha=1\,\,\textrm{for all}\,\, g\inG \}$. In this paper, we state some basic properties of the graph, like connectivity, diameter, girth and Hamiltonian. Moreover, planarity and 1-planarity are also investigated here.

References

  • A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006), 468-492.
  • A. Abdollahi, Engel graph associated with a group, J. Algebra, 318 (2007), 680-691.
  • A. Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group, Comm. in Algebra, 35 (2007), 2057-2081.
  • R. Barzegar and A. Erfanian, Nilpotency and solubility of groups relative to an automorphism, Caspian Journal of Mathematical Sciences, 4(2) (2015), 271-283.
  • A. Erfanian, M. Farrokhi D.G. and B. Tolue, Non-normal graphs of finite groups, J. Algebra Appl., 12 (2013).
  • J. A. Bondy and J. S. R. Murty, Graph Theory with Applications, Elsevier, (1977).
  • P. J. Cameron and S. Ghosh, The power graph of a fnite group, Discrete Math, 311 (2011), 1220-1222.
  • I. Fabrici and T. Madaras, The structure of 1-planar graphs, Discrete Mathematics, 307 (2007), 854 - 865.
  • A. Iranmanesh and A. Jafarzadeh, On the commuting graph associated with the symmetric and alternating groups, J. Algebra Appl., 7 (2008), 129-146.
  • D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag: New York-Heidelberg Berlin (1982).
  • The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.6.4, 2013 (http://www.gap-system.org/).
  • J. S. Williams, Prime graph components of finite groups, J. Algebra, 69 (2) (1981), 487-513.
There are 12 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

M. Mahtabi This is me

A. Erfanian

Publication Date February 1, 2018
Published in Issue Year 2018 Volume: 47 Issue: 1

Cite

APA Mahtabi, M., & Erfanian, A. (2018). A graph associated to a fixed automorphism of a finite group. Hacettepe Journal of Mathematics and Statistics, 47(1), 93-99.
AMA Mahtabi M, Erfanian A. A graph associated to a fixed automorphism of a finite group. Hacettepe Journal of Mathematics and Statistics. February 2018;47(1):93-99.
Chicago Mahtabi, M., and A. Erfanian. “A Graph Associated to a Fixed Automorphism of a Finite Group”. Hacettepe Journal of Mathematics and Statistics 47, no. 1 (February 2018): 93-99.
EndNote Mahtabi M, Erfanian A (February 1, 2018) A graph associated to a fixed automorphism of a finite group. Hacettepe Journal of Mathematics and Statistics 47 1 93–99.
IEEE M. Mahtabi and A. Erfanian, “A graph associated to a fixed automorphism of a finite group”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 1, pp. 93–99, 2018.
ISNAD Mahtabi, M. - Erfanian, A. “A Graph Associated to a Fixed Automorphism of a Finite Group”. Hacettepe Journal of Mathematics and Statistics 47/1 (February 2018), 93-99.
JAMA Mahtabi M, Erfanian A. A graph associated to a fixed automorphism of a finite group. Hacettepe Journal of Mathematics and Statistics. 2018;47:93–99.
MLA Mahtabi, M. and A. Erfanian. “A Graph Associated to a Fixed Automorphism of a Finite Group”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 1, 2018, pp. 93-99.
Vancouver Mahtabi M, Erfanian A. A graph associated to a fixed automorphism of a finite group. Hacettepe Journal of Mathematics and Statistics. 2018;47(1):93-9.