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

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  1. The paper classifies finite non-abelian groups based on the embeddability of their commuting and non-commuting graphs.
  2. Commuting graphs of finite groups can only be planar, toroidal, or projective under specific conditions.
  3. The genus and crosscap number are crucial for determining the embeddability of graphs on surfaces.
  4. Only groups S3, D8, and Q8 have planar non-commuting graphs, according to Theorem 3.1.
  5. No finite groups exhibit toroidal non-commuting graphs, as established in Theorem 3.2.

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References (13)

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  13. Department of Mathematics, University of Neyshabur, P.O.Box 91136-899, Neyshabur, Iran E-mail address: mojgan.afkhami@yahoo.com