There are several kind of graphs attached to finite groups, such as conjugacy graphs, degree graphs and character graphs. In this thesis, we shall focus on character graphs. The character graph (G) of a finite group G is defined in the following way. The vertices of the graph are the nonlinear complex irreducible characters of G, and there is an edge between two vertices and if and only if (1) and (1) have a common prime divisor. Let G be a finite group. Irr(G) is the set of its complex irreducible character and Irr?(G) is the subset of Irr(G) consisting of the nonlinear ones. If three elements of Irr?(G) have common prime divisor then the character graph of G has a triangle. If G is an abelian group, then the graph of G is empty. If G is nonabelian, according to type="#_x0000_t75" A?, the alternating group on five letters. According to the above theorem, we can get that, for a finite group, its character graph has no triangles if and only if the graph dosen’t contain a cycle, i.e. each connected component of the graph is a tree. First, we use the classification theorem of finite simple groups and we show that A? is the only nonabelian simple group whose character graph contains no triangles. Then we prove that a group must be isomorphism to A? if it is perfect and its character graph has no triangles. In addition, we argue that a group must perfect if it is nonsolvable and its character graph dosen’t have a triangle. Finally, the main theorem follows from the above results.