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The twin non-commuting graph of a group. (English) Zbl 1442.05091

Summary: In this paper, the twin non-commuting graph of the group \(G\) is introduced by partitioning the non-commuting graph vertices. This classification of the vertices is based on a special property, namely the twin vertices property. We choose a vertex from each class as a representing vertex for the twin non-commuting graph. Moreover the adjacency of two vertices in a twin non-commuting graph depends on the connectivity of them in the main non-commuting graph. We observe that the twin non-commuting graph of an AC-group is a complete graph. Moreover, some results about the metric dimension of the non-commuting graph are obtained. For instance, the metric dimension of the non-commuting graph is 3 if and only if it is associated with \(S_3\), \(D_8\), \(Q_8\).

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C40 Connectivity
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Software:

GAP
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References:

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