Character degree graphs

Character Theory has an important place in understanding the structures of finite groups. There are many studies in the literature on the relationships between the structure of a group and its irreducible characters. For example, it was proved in 1904 by Burnside using Character Theory that the groups of order p^a q^b are solvable, where p and q are prime numbers. In this thesis, after giving place to the basic definitions and theorems in Group Theory and Character Theory, the definition and properties of character degree graph are given. Pálfy's Three Vertices Theorem for character degree graphs of solvable groups is explained in detail. Lewis has given a complete classification of solvable groups whose character degree graphs are disconnected. Then Lewis and White have discribed the structure of nonsolvable groups whose character degree graphs are disconnected. These studies are mentioned in the thesis. In the part of results, a complete classification of all rational groups whose character degree graphs are disconnected is given.