标题： 图的全支配多项式 显示英文标题

标题： Total domination polynomials of graphs

摘要： 给定一个图$G$，一个全支配集$D_t$是一个顶点集合，使得$G$的每个顶点都与$D_t$的某些顶点相邻，并令$d_t(G,i)$为大小为$i$的所有全支配集的数量。 全支配多项式，定义为$D_t(G,x)=\sum\limits_{i=1}^{| V(G)|} d_t(G,i)x^i$，最近已成为支配理论领域中备受关注的扩展研究之一。 在本文中，我们得到了全支配多项式的顶点约化和边约化公式。 作为结果，我们给出了路径和环的全支配多项式。 此外，我们确定了树的全支配多项式的尖锐上界，并刻画了达到这些上界的相应图。 最后，我们使用约简公式来研究$G$中顶点集和全支配多项式之间的关系。 显示英文摘要

摘要： Given a graph $G$, a total dominating set $D_t$ is a vertex set that every vertex of $G$ is adjacent to some vertices of $D_t$ and let $d_t(G,i)$ be the number of all total dominating sets with size $i$. The total domination polynomial, defined as $D_t(G,x)=\sum\limits_{i=1}^{| V(G)|} d_t(G,i)x^i$, recently has been one of the considerable extended research in the field of domination theory. In this paper, we obtain the vertex-reduction and edge-reduction formulas of total domination polynomials. As consequences, we give the total domination polynomials for paths and cycles. Additionally, we determine the sharp upper bounds of total domination polynomials for trees and characterize the corresponding graphs attaining such bounds. Finally, we use the reduction-formulas to investigate the relations between vertex sets and total domination polynomials in $G$.