Abstract: To study the unimodality of stable set partitions, the concept of its adjoint polynomials is proposed and the adjoint polynomials of several graphs are obtained. Using the adjoint polynomials, the unimodality of various graphs is demonstrated, including star graphs, complete products of star graphs, complete q-partite graphs,（ n-2） - regular graphs, complete products of（ n-2） - regular graphs, etc. Furthermore, the number of all possible partitions of stable sets is calculated skillfully. Finally, a generalization of the unimodal theorem is extended, and several classical examples are derived. In addition, a generalization of one form of the Newton inequalities is given, and a series of related inequalities are obtained.

Key words:  , stable set, partition, adjoint polynomial, complete product, unimodality