The non-commuting graph of a non-Abelian group is defined as follows. The vertex set of is and two vertices and are joined by an edge if and only if . This graph was first defined by P. Erdős which is quoted by 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 are related to the group theoretical properties of .
In the paper under review the authors answer some questions about and relate them to the structure of . But the bulk of the paper is centered around the verification of the following Conjecture: Let and be two non-Abelian groups with the property that and are isomorphic graphs, then , and if is a simple group .
The authors prove the first part of the conjecture for the groups , , , or a non-solvable AC-group, and the second part for the groups and the Suzuki groups , . Some invariants of the graph , such as the clique number, chromatic number, etc., are found for special groups .