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Commuting graphs are characterized by vertices that are non-central elements of a group where two vertices are adjacent when they commute. In this paper, the concept of commuting graph is extended by defining the generalized commuting graph. Furthermore, the generalized commuting graph of the dihedral groups, the quasi-dihedral groups and the semi-dihedral groups are presented and discussed. The graph properties including chromatic and clique numbers are also explored.

Generalized Commuting Graph of Dihedral, Semi-dihedral and Quasi-dihedral Groups