: In this thesis we deal with wreath product of finite simple groups and cyclic groups. We investigate the basic properties of wreath products. Also we present 0cm 0cm 10pt" For a finite group G let \\Gamma(G) denote the graph definde on the nonidentity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this thesis we study some properties the generating graph of monolitic groups.We prove that for a monolithic group G with non-Abelian socle N such that G\ is a cyclic group, if \\Gamma(G) contains a Hamiltonian cycle, then \\Gamma(G) is pancyclic provided that cardinal N is large enough. Also we prove that if m is odd and the number of prime numbers dividing m is at most 140, then there exists a positive integer \au such that if S is a simple group of Lie type and , then the graph \\Gamma(S\\wr C-{m}) contains a Hamiltonian cycle.