Some Structural Properties of Power and Commuting Graphs Associated With Finite Groups

‎In this talk we will investigate some properties of the power and commuting graphs associated with‎ ‎finite groups‎, ‎using their tree-numbers‎. ‎It has been shown that the simple group‎ ‎$L_2(7)$ can be characterized through the tree-number of its power graph‎, ‎and the classification‎ ‎of groups with a power-free decomposition is presented‎. ‎We have also obtained‎ ‎an explicit formula concerning the tree-number of commuting graphs‎ ‎associated with the Suzuki simple groups‎