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Non-commuting graph of a group. (English) Zbl 1105.20016

The non-commuting graph Γ G of a non-Abelian group G is defined as follows. The vertex set of Γ G is V(G)=G-Z(G) and two vertices x and y are joined by an edge if and only if xyyx. 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 Γ G are related to the group theoretical properties of G.

In the paper under review the authors answer some questions about Γ 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 Γ G and Γ H are isomorphic graphs, then |G|=|H|, and if G is a simple group GH.

The authors prove the first part of the conjecture for the groups GS n , A n , PSL(2,q), D n or a non-solvable AC-group, and the second part for the groups GPSL(2,2 n ) and the Suzuki groups 2 B 2 (2 2n+1 ), n>1. Some invariants of the graph Γ G , such as the clique number, chromatic number, etc., are found for special groups G.


MSC:
20D60Arithmetic and combinatorial problems on finite groups
05C25Graphs and abstract algebra
20D06Simple groups: alternating groups and groups of Lie type