A note on the coprime graph of a group

Document Type: Research Paper

Author

Department of Basic Sciences, University of Jiroft, Jiroft, Kerman, Iran

Abstract

‎In this paper we study the coprime graph of a group $G$‎. ‎The‎ ‎coprime graph of a group $G$‎, ‎denoted by $\Gamma_G$‎, ‎is a graph‎ ‎whose vertices are elements of $G$ and two distinct vertices $x$‎ ‎and $y$ are adjacent if and only if $(|x|,|y|)=1$‎. ‎In this paper‎, ‎we ‎show that $\chi(\Gamma_G)=\omega(\Gamma_G).$ We classify all the‎ ‎groups which $\Gamma_G$ is a complete $r-$partite graph or a‎ ‎planar graph‎. ‎Also we study the automorphism group of‎ ‎$\Gamma_G$‎.

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