A SPECIAL CLASS OF GRAPHS EMERGING FROM QUATERNION GROUPS

This work explores new graphs constructed on generalized Quaternion groups . The vertices are the conjugacy classes of and there will be an edge between the two conjugacy classes if their orders are distinct. We define this simple graph as order structured conjugacy class simple graph (OSCSG), labelled by . In this research, we characterize the proposed graph and prove that for , the closure of can never be a complete graph. Its complement, always contains a maximum complete subgraph for . Moreover, for , the upper bound of average distance is determined. Finally, the bipartite and tri-partite nature of these graphs along with their spectral characteristics is also discussed.