A study on some graphs associated with finite groups / Lim Ming Chyang - Lim , Ming Chyang (2019)
In this dissertation, we study properties of various graphs associated with finite groups and investigate the extent to which these graphs determine the groups. Among the graphs associated with finite groups that we consider here are conjugate graphs, order graphs and generalised order graphs. For conjugate graphs and order graphs associated with certain groups, we determine the number of complete components in the graphs and their clique numbers. The main focus of this research is on generalised order graphs of finite groups. By studying relationships between power graphs and generalised order graphs, we prove that for a finite group, k-connectedness of its power graph implies k-connectedness of its generalised order graph. We also prove that the generalised order graph and the power graph associated with a finite cyclic group are isomorphic. In addition, we classify certain classes of finite groups according to various graph properties of the associated generalised order graphs. We also prove that the generalised order graph of a finite abelian group is 3-connected and Hamiltonian. Along the way we also prove some number-theoretic inequalities.