全控制多项式的零点分布

Distribution of roots of total domination polynomial

n阶图G的全控制多项式定义为dt(G,x)=∑i=γt(G)d^n t(G,i)x^i,其中dt(G,i)是G的尺寸为i的全控制集的数目,γt(G)是G的全控制数.首先借助Beraha-Kahane-Weiss定理给出friendship图的全控制多项式根的极限曲线,然后证明了控制多项式的根在整个复平面上是稠密的.

The total domination polynomial of a graph G of order n is the polynomial dt(G,x)=∑i=γt(G)d^n t(G,i)x^i,where d t(G,i)is the number of total dominating sets of G of size i,andγt(G)is the total domination number of G.In this paper,we first show the limiting curves for the total domination roots of friendship graphs using the Beraha-Kahane-Weiss Theorem.Then we show that the total domination roots are dense in the entire complex plane.