树的稳定子集和稳定指标

一个实矩阵的符号稳定性问题在经济学、生态学等诸多领域中都有应用背景．本文利用1]中给出的不可约矩阵的符号稳定性的有关结论，将一个实矩阵的符号稳定性判定问题转化为一个等价的图论问题，即判定无向树中一个点子集的稳定性问题．本文引入了树的稳定子集的概念并给出了稳定子集的递归判别方法．本文还提出井研究了树的稳定指标，即树中所有稳定子集的最小基数，证明了关于稳定指标的一个min—max型定理，井给出了n阶树的稳定指标的最好上界及达到上界的极树的完全刻划。

ON STABLE SUBSETS AND STABLE INDICES OF TREES

The study of sign stability of a real square matrix has important applications in various areas such as economics, ecology, and so on. In this paper, the sign stability problem for matrices is turned into an equivalent graph theoretical problem, namely, the stability problem for the vertex subsets of trees. We introduce the concept of stable vertex subsets of trees and give a recursive method for recognizing them. We also introduce for trees a new parameter, stable index, which is the smallest cardinality among all the stable vertex subsets of the tree. We prove a min-max type theorem for stable index, give a sharp upper bound for the stable indices of trees of order n, and give a complete characterization for the extreme trees whose stable indices attaining this upper bound.