Toroidal and projective commuting and non-commuting graphs (original) (raw)
Abstract
In this paper, all finite groups whose commuting (non-commuting) graphs can be embed on the plane, torus or projective plane are classified.
Key takeaways
AI
- The paper classifies finite non-abelian groups based on the embeddability of their commuting and non-commuting graphs.
- Commuting graphs of finite groups can only be planar, toroidal, or projective under specific conditions.
- The genus and crosscap number are crucial for determining the embeddability of graphs on surfaces.
- Only groups S3, D8, and Q8 have planar non-commuting graphs, according to Theorem 3.1.
- No finite groups exhibit toroidal non-commuting graphs, as established in Theorem 3.2.
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References (13)
- A. Abdollahi, S. Akbari and H.R. Maimani, Non-commuting graph of a group, J. Algebra 298 (2006), 468-492.
- J. Battle, F. Harary, Y. Kodama and J. W. T. Youngs, Additivity of the genus of a graph, Bull. Amer. Math. Soc. 68 (1962), 656-568.
- L.W. Beineke and F. Harary, Inequalities involving the genus of a graph and its thickness, Prcoc. Glasgow Math. Assoc. 7 (1965), 19-21.
- E.D. Bolker, Groups whose elements are of order two or three, Amer. Math. Monthly 79(9) (1972), 1007-1010.
- A. Bouchet, Orientable and nonorientable genus of the complete bipartite graph, J. Combin. Theory Ser. B 24 (1978), 24-33.
- The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.6.4, 2013 (http://www.gap-system.org/).
- H.H. Glover, J.P. Huneke and C.S. Wang, 103 Graphs that are irreducible for the projective plane, J. Combin. Theory Ser. B 27 (1979), 332-370.
- W.H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80 (1973), 1031-1304.
- K. Kuratowski, Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 271-283.
- W. Massey, Algebraic Topology: An Introduction, Harcourt, Brace & World, Inc., New York, 1967.
- G. Ringel, Map Color Theorem, Springer-Verlag, New York, Heidelberg, 1974.
- R. M. Solomon and A. J. Woldar, Simple groups are characterized by their non-commuting graphs, J. Group Theory 16(6) (2013), 793-824.
- Department of Mathematics, University of Neyshabur, P.O.Box 91136-899, Neyshabur, Iran E-mail address: mojgan.afkhami@yahoo.com