基于电阻距离的三类Kirchhoff型指标研究

Kirchhoff index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index are important topological indices based on resistance distance in Chemical Graph Theory, and the problems about them are all frontier and hot topics. By means of graph theory, algebra and probability, this project plans to give a systematic research on the three Kirchhoffian indices. Firstly, we research the connected graphs with maximum and minimum the three Kirchhoffian indices when different parameters are respectively given. Secondly, we research the unicyclic graphs with maximum and minimum the three Kirchhoffian indices when different parameters are respectively given, and with the first several maximum additive degree-Kirchhoff indices. Thirdly, we research the bicyclic graphs and other special graphs including cycles with maximum and minimum the three Kirchhoffian indices when different parameters are respectively given, and with the first several maximum and minimum the two degree-Kirchhoff indices. Finally, we research the relation between the three Kirchhoffian indices and some classical topological indices, and the properties and applications of the three Kirchhoffian indices in some complex network graphs. The research results of the above problems have not only important theoretical significance in Mathematics，but also applicable value in Chemical and Physics.

基于电阻距离的Kirchhoff指标、multiplicative degree-Kirchhoff指标和additive degree-Kirchhoff指标是化学图论中的重要拓扑指标，所涉及的问题均是前沿热点课题。本项目拟运用图论、代数和概率方法研究：(1)连通图分别给定不同参数时三类Kirchhoff型指标的极值和极图；(2)单圈图分别给定不同参数时三类Kirchhoff型指标的极值和极图，及具有前几大additive degree-Kirchhoff指标的极值和极图；(3)双圈图及其它特殊含圈图类分别给定不同参数时三类Kirchhoff型指标的极值和极图，及具有前几大(小)两类degree-Kirchhoff指标的极值和极图；(4)三类Kirchhoff型指标与多种经典指标间的关系，及在多种复杂网络中的性质和应用。以上问题的研究具有重要的数学理论意义和化学物理等应用价值。

图参数是图论中重要的研究内容。拓扑指标是定量描述分子结构的图参数，是研究分子图的重要工具。本项目主要研究基于电阻距离的三类Kirchhoff指标极值和极图问题，以及基于最短路距离的拓扑指标极值和极图问题。本项目取得的主要研究结果有：运用统一方法确定三类Kirchhoff型指标在给定阶数和最大度单圈图中的最大极值和极图；确定multiplicative degree-Kirchhoff指标在给定阶数单圈图中前七大极值排序和极图；确定additive degree-Kirchhoff指标在给定阶数单圈图中前十大极值排序和极图；确定additive degree-Kirchhoff指标在给定阶数和圈数仙人掌图中最大最小极值和极图；运用通用方法确定两类离心率指标在给定阶数的树、单圈图和双圈图中极值排序和极图；确定Szeged指标在给定阶数全负荷单圈图中前三大和前三小极值排序和极图；确定几类哈密顿连通图的Detour指标。项目组目前已发表5篇SCI学术论文，研究成果不仅推广和丰富了图论相关问题的研究内容，而且在化学、物理和网络科学等领域具有广泛的应用背景，对科学技术和经济建设的发展将起到良好的推动作用。