一类网络拓扑性质的研究

Research on the Topological Properties of a Class of Networks

摘要: 当今社会，复杂网络无处不在，网络的拓扑特性与诸多有趣问题紧密联系，首先，介绍了一类环形网络的构造过程，接着，深入探究了这类网络的拓扑特性及其生成树的枚举问题。给出了这类网络的拓扑性质及其生成树数目公式的确解。在此基础上，进一步推导了这类网络进行边收缩后其变形网络的拓扑性质。最后，任意边赋权下该类环形网络的最小生成树问题，提出了一种算法，能够在任意条件下确定最小生成树的结构。在实际应用的背景下，研究了小规模该网络最小生成树的结构问题，采用Python进行了算法运行，验证了该算法的实用性与有效性，为实际应用提供了理论支持与实践依据。

Abstract: In today’s society, complex networks are ubiquitous, and the topological properties of networks are closely intertwined with many intriguing problems. Firstly, the construction process of a class of ring networks is introduced. Subsequently, the topological properties of this class of networks and the enumeration problem of their spanning trees are investigated in depth. The exact solutions for the topological properties of this class of networks and the formula for the number of spanning trees are derived. On this basis, the topological properties of the deformed networks obtained by edge contraction of this class of networks are further deduced. Finally, for the minimum spanning tree (MST) problem of this class of ring networks under arbitrary edge weight assignments, an algorithm is proposed, which is capable of determining the structure of the MST under any conditions. In the context of practical applications, the structural problem of the MST for small-scale networks of this type is studied, and the algorithm is implemented using Python. The practicability and effectiveness of the algorithm are verified, providing theoretical support and practical basis for real-world applications.