乘积图生成树的Wiener数问题

On the Wiener Index of Cartesian Product Graphs

研究了给定一个连通图,如何确定其Wiener数最小的生成树问题。Dobrynin等构造了超立方体的两类Wiener数“很小”的生成树,并进一步猜想这两类树都是Wiener数最小的生成树。利用归纳推理及递归关系,对更一般的且具有良好拓扑性质和较高网络模型应用价值的乘积图,如G1×G2、Kmn等,构造了相应的生成树并计算了它们的Wiener数的值,以期获得这些乘积图Wiener数最小的生成树。这些结果推广了Dobrynin关于超立方体的结果。

This paper focuses on the problem： finding a spanning tree of a graph with minimum Wiener number. Dobrynin constructed two types of spanning trees for hypercube and further conjectured that these trees are the minimum spanning trees （with respect to the Wiener number）. In this paper, we construct some spanning trees for more general graphs： Cartesian product graphs （known with good topological properties and excellent network parameter） such as G1×G2、Kn^m. etc., in an attempt to find their minimum spanning trees. Our result generalizes that of Dobrynin＇s. In our study, the mathematical in duction and recurrence relation techniques will play important roles.