稳定集合分拆的伴随多项式和单峰性

Adjoint Polynomials and Unimodality of Stable Set Partitons

为了研究稳定集合分拆的单峰性,提出伴随多项式的概念,并获得了几个图的伴随多项式。利用伴随多项式,证明了多种图的单峰性,如星形图、星形图的完全积、完全q-部图、(n-2)-正则图和(n-2)-正则图的完全积等。进一步,技巧性地计算了稳定集合分拆的所有个数。最后,推广了单峰性定理,并且导出几个例子,同时还推广了牛顿不等式的形式,得到了系列不等式。

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.