Metric Properties of Non-Commuting Graph Associated to Two Groups

The non-commuting graph associated to a group has non-central elements of the graph as vertices and two elements x and y do not form an edge if and only if xy = yx. In this paper, we consider non-commuting graphs associated to dihedral and semidihedral groups. We investigate their metric properties such as center, periphery, eccentric graph, closure and interior. We also perform various types of metric identifications on these graphs. Moreover, we generate metric and metric-degree polynomials of these graphs.