有限网络的一个性质

An Interesting Property of Finite Networks

借助实例提出一个有关有限网络研究的基本问题:对装置的任意电路网络是否总能经有限次改变状态后,从全"关闭"状态变为全"开启"状态.利用无向图作为网络的数学模型,并利用邻接矩阵,可把该基本问题归结为证明二元域Z2上一个特殊的线性方程组是否有解的问题.基于二元域Z2的运算性质及此域上线性方程组的理论,可严格证明上述线性方程组解的存在性,进而可证明基本问题存在肯定性解答.

In this paper we give an example to present a basic problem tor finite networks： whether it is possible or not for any finite network of devices to change its status from the initial status with all its devices being in ＂close＂ state to the final status with all its devices being in ＂open＂ state？ An undirected graph whose adjacency matrix is properly defined can be used as a mathematical model of the network. We confirm the positive answer for the basic problem by proving that the corresponding system Of linear equations is consistent in the field Z_2.