图中距离参数的研究

The distance problem of graphs has a wide range of applications in real life, for instance, research on network optimization,data transmission, molecular structure etc are closely related with the distance of a grpah. Many attentions are paied to the relationship between those parameters, such as average distance, diameter and radius, and the others. There are still many conjectures and open problems which have not been solved. Dankelmann recently proposed the concept of weighted average distance, and presented some interesting conjectures and problems. We will focus on these conjectures and problems, and strive to give a complete or part solution to them.

图的距离问题在实际生活中有着广泛的应用，例如网络优化，数据传输，分子结构等的研究与距离有着紧密的联系。图的距离相关的参数, 如平均距离、直径、半径与图的其它参数之间的关系一直受到很多图论学者的关注和研究。这一领域中，仍有很多猜想和公开问题还没有解决。Dankelmann最近给出了赋权平均距离的概念，并提出了一些有趣的猜想与问题。我们将围绕这些猜想和问题展开研究，力求完全或部分地对它们给出回答。

图的距离问题的研究一直受到图论学者的关注，关于图的距离的相关参数的研究中有很多猜想和公开问题至今还没有解决。本项目主要研究了与距离相关的Wiener指标，我们确定了线图的补图的Wiener指标。图的Steiner距离是图的距离概念的推广，Steiner k-Wiener指标是Wiener指标概念的推广。我们主要研究了平方图的Steiner距离，以及Steiner k-Wiener指标，对于k=3时，给出了平方图的Steiner k-Wiener指标的Nordhaus-Gaddum型关系。对于给定二部划分的树，确定了其Steiner k-Wiener指标的上下界，给出了相应的极图。