A Kind of Non-Commuting Graph Associated with the Derived Subgroup and Disjoint Two-Element Subsets of a Group

This paper introduces a new graph associated with a subgroup of a finite group, whose vertices are the two-element subsets of the subgroup satisfying the prescribed condition, while adjacency is defined by a new relation between these subsets. The proposed graph provides a novel combinatorial model for studying subgroup structures from the perspective of algebraic graph theory. A connection between the proposed graph and the -noncommuting graph introduced by Olue, Erfanian, and Jafarzadeh is established by proving that the connectivity of the proposed graph implies the connectivity of the corresponding -noncommuting graph, whereas the converse does not hold in general. A counterexample is presented to demonstrate this fact. Finally, the proposed graph is applied to the dihedral group, and its main graph-theoretic properties are determined.