Let $G$ be a non-abelian group with center $Z(G)$ . Associate a graph $\Gamma_G$ with $G$ whose vertices are the non-central elements $G\setminus Z(G)$ and whose edges join those vertices $x,y\in G\setminus Z(G)$ for which $xy\neq yx$ . Then $\Gamma_G$ is said to be the non-commuting graph of $G$ . The non-commuting graph $\Gamma_G$ was first considered by Paul Erdös, when he posed the following problem in 1975 [B.H. Neumann, A problem of Paul Erdös on groups, J. Austral. Math. Soc. Ser. A 21 (1976), 467-472]:
Let $G$ be a group whose non-commuting graph has no infinite complete subgraph. Is it true that there is a finite bound on the cardinalities of complete subgraphs of $\Gamma_G$ ?
B. H. Neumann answered positively Erdös' question.
The non-commuting graph $\Gamma_G$ of a non-abelian group $G$ is always connected with diameter 2 and girth 3. It is also Hamiltonian. $\Gamma_G$ is planar if and only if $G$ is isomphic to the symmetric group $S_3$ , or the dihedral group $\mc{D}_8$ of order $8$ or the quaternion group $Q_8$ of order $8$ .
See [Alireza Abdollahi, S. Akbari and H.R. Maimani, Non-commuting graph of a group, Journal of Algbera, 298 (2006) 468-492.] for proofs of these properties of $\Gamma_G$ .

Examples

Symmetric group $S_3$

Figure 1: Non-commuting graph of the symmetric group $S_3$

Octic group

Figure 2: Non-commuting graph of the octic group