A self-cetered graph G is said to be complementary and self-centered graph if its complement (?) also is self-centered graph. A complementary and self-centered graph G with G≌(?) is called as self-complementary and self-centered graph. This paper deals with some properties of complementary and self-centered graph. A generalized method of constructing self-complementary and self-centered graph is given. Thus a way of constructing self-centered graph is provided.