## Abstract

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 based on special property, precisely being 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 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 is obtained. For instance, the metric dimension of the non-commuting graph is 3 if and only if it is associated with \(S_3,D_8,Q_8\).

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Tolue, B. The twin non-commuting graph of a group.
*Rend. Circ. Mat. Palermo, II. Ser* **69, **591–599 (2020). https://doi.org/10.1007/s12215-019-00421-4

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### Keywords

- Non-commuting graph
- Twin graph
- Metric dimension

### Mathematics Subject Classification

- 05C25
- MSC 20P05