For a finite group , we denote by the set of element orders of . Let , where is the number of elements of order in . In this thesis we show that and where and is a prime number , are uniquely determined by . Also we define the common divisor graph associated with and denote by . We study some parameters of uch as diameter , the number of components and the automorphism graph and planarity of graph. We study groups with regular graphs . Also we determine the structure of groups with acyclic graphs .