Let G be a finite group. The character graph ?(G), where the characters are over an algebraically closed field of characteristic zero, is defined as follows: the set of vertices is the set of all nonlinear irreducible characters of G, and vertice ? and ? are joined by an edge if and only if ?(1) and ?(1) have a common prime divisor. So, G is abelian if and only if ?(G) has no vertices. In this thesis first, non-nilpotent groups with two irreducible character degrees are characterized. This is done by using of a description of solvable groups in which the commutator subgroup is a minimal normal subgroup. Then, all the finite solvable groups G whose character graph ?(G) have no triangles are at most two non-linear irreducible characters and the symmetric group 4 .