Some results on characterization of finite group by non commuting graph

Document Type: Research Paper

Authors

1 University of Tehran

2 K. N. Toosi University of Technology

Abstract

The non commuting graph $\nabla(G)$ of‎ ‎a non-abelian finite group $G$ is defined as follows‎: ‎its vertex set is‎ ‎$G‎- ‎Z (G)$ and two distinct vertices $x$ and $y$‎ ‎are joined by an edge if and only if the commutator of $x$ and $y$ is not the‎ ‎identity‎. ‎In this paper we prove some new results about this graph‎. ‎In‎ ‎particular we will give a new proof of Theorem 3.24 of [A‎. ‎Abdollahi‎, ‎S‎. ‎Akbari‎, ‎H‎. ‎R‎, ‎Maimani‎, ‎Non-commuting graph of a group‎, ‎J‎. ‎Algebra‎, ‎298 (2006) 468-492.]‎. ‎We also prove that‎ ‎if $G_1‎, ‎G_2‎, ‎\ldots‎, ‎G_n$ are finite groups such that $Z(G_i)=1$ for $i=1‎, ‎2‎,‎\ldots‎, ‎n$‎ ‎and they are characterizable by non commuting graph‎, ‎then $G_1 \times G_2‎ ‎\times \cdots \times G_n$ is characterizable by non-commuting graph‎.

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A. Abdollahi, S. Akbari, H. R. Maimani (2006). Non-commuting graph of a group. J. Algebra. 298, 468-492
M. R. Darafsheh (2009). Groups with the same non-commuting graph. Discrete Appl. Math.. 157 (4), 833-837
Ron Solomon and Andrew Woldar (2012). All Simple groups are characterized by their non-commuting graphs. preprint.