The non-commuting graph \(\Gamma_G\) of a non-Abelian group \(G\) is defined as follows. The vertex set of \(\Gamma_G\) is \(V(G)=G-Z(G)\) and two vertices \(x\) and \(y\) are joined by an edge if and only if \(xy\neq yx\). This graph was first defined by P. Erdős which is quoted by \textit{B. H. Neumann} [J. Aust. Math. Soc., Ser. A 21, 467-472 (1976; Zbl 0333.05110)]. A natural question to ask is how the graph theoretical properties of \(\Gamma_G\) are related to the group theoretical properties of \(G\). In the paper under review the authors answer some questions about \(\Gamma_G\) and relate them to the structure of \(G\). But the bulk of the paper is centered around the verification of the following Conjecture: Let \(G\) and \(H\) be two non-Abelian groups with the property that \(\Gamma_G \) and \(\Gamma_H\) are isomorphic graphs, then \(|G|=|H|\), and if \(G\) is a simple group \(G\cong H\). The authors prove the first part of the conjecture for the groups \(G\cong S_n\), \(A_n\), \(\text{PSL}(2,q)\), \(D_n\) or a non-solvable AC-group, and the second part for the groups \(G\cong\text{PSL}(2,2^n)\) and the Suzuki groups \(^2B_2(2^{2n+1})\), \(n>1\). Some invariants of the graph \(\Gamma_G\), such as the clique number, chromatic number, etc., are found for special groups \(G\).