图的基于距离的拓扑指标及若干相关问题

The research on various topological indices of graphs is a hot topic in chemical graph theory. We will study on several topological indices of graphs which are based on distance in graph, such as Wiener index, hyper-Wiener index, Harary index and Kirchhoff index which are extensively studied, and eccentric distance sum (EDS) which is recently studied，etc. Zagreb indices are also distance-based topological indices according to the newest researching result. These indices of this kind have some nice mathematical properties and been found some important applications in chemical graph theory. It is fundamental from theoretical and applicable viewpoints to determine the bounds for these topological indices of graphs from some given set and characterize the corresponding extremal graph. There are some other important directions in chemical graph theory, which are to study the inner correlation among several topological indices of graphs and to establish some numerical relationship among them, and based on it to construct some unified approach to determine the extremal graphs in some given set with respect to several topological indices as well as the inverse problem for topological indices. In this research item, based on some excellent known results of this type, we will explore the numerical relation among several distance-based topological indices as well as determine the extremal graphs among ones with given parameter with respect to topological indices of this kind. Also, in this item, some unified approach will be invented to extremal graph from some given set with respect to as more topological indices of this kind as possible, moreover we will make a further study on the inverse problem for topological indices of this kind.

图的各种拓扑指标的研究是化学图论中的热点问题。我们拟研究图的基于距离的几类拓扑指标，如广泛研究的Wiener 指标和hyper-Wiener 指标、Harary 指标、Kirchhoff指标及最近研究的偏心距离和（EDS）等。根据最新研究，图的Zagreb 指标也属于此类拓扑指标。这类指标在化学图论中有着重要的应用，具有很好的数学性质。确定给定图类中关于此类拓扑指标上下界，刻画相应的极图有着深刻的理论和实际意义。研究此类拓扑指标之间的内在联系，确定其数量关系，以及在此基础上，构造给定图类中关于某几种拓扑指标的极图的统一方法及关于拓扑指标的逆问题，都是化学图论中的重要方向。本项目在分析同类研究的基础上，确定给定参数下关于此类拓扑指标的极图，探究几种拓扑指标之间的数量关系，构造给定图类中关于尽可能多的拓扑指标的极图的统一方法，我们还将在关于此类拓扑指标的逆问题上做一些深入探讨。

基于距离的拓扑指标是图的重要不变量之一，在化学图论中有着重要而广泛的应用。本项目中，我们主要研究基于距离的拓扑指标，包括的极值问题，这类指标与图的其他不变量的内在关系等，基于顶点度的不变量（特定条件下也属于距离不变量）的数学性质等。该项目资助下，我们研究了k-apex树的加权Harary指标的极值性质，确定了各种条件下关于逆度的极图。我们还分别刻画了具有m条边的n阶连通图的Zagreb指标及基于距离度的不变量之间的关系，利用图的矩阵，获得了图的基于广义距离（电阻距离）不变量的数学结论。此外，我们出版了一篇关于图的基于距离的拓扑指标的综述论文和一部关于图的Harary指标的英文专著一部。该项目取得的成果在图论研究中具有重要的理论意义，会在很大程度上推动的图论研究的发展。