化学图的谱及相关性质

A chemical structure can be conveniently represented by a graph, which is called a chemical graph or a molecular graph. Chemical graph theory is the topology branch of mathematical chemistry which applies graph theory to mathematical modelling of chemical phenomena. Graph theory is used to mathematically model molecules in order to gain insight into the physical properties of these chemical compounds. Some physical properties, such as the boiling point, are related to the geometric structure of the compound. This is especially true in the case of chemical compounds known as alkanes. Alkanes are organic compounds exclusively composed of carbon and hydrogen atoms. Hexagonal systems are of great importance for theoretical chemistry because they are the natural graph representation of benzenoid hydrocarbons. A considerable a amount of research in mathematical chemistry has been devoted to hexagonal systems. But it is very difficult to determine the spectra of hexagonal graphs. This project mainly puts emphasis on calculating the spectra of chemical graphs and hexagonal systems. This may help to estimate the physical properties of the chemical compounds.

化学结构可以很简单地表示成图的形式，这样的图也称为化学图，或者分子图。对化学图的研究是图论的重要应用之一。通过对化学图的某些不变量的研究，可以反映相应的化合物的一些物理性质比如沸点、熔点等。目前这个领域的研究主要是针对烷烃类的化合物，即碳氢化合物。化学图中很重要的一种是所谓的六角系统图。这种图是苯环碳氢化合物的图论表示。但是要完全计算出某种六角系统的谱是十分困难的。本项目希望计算一些特殊的化学图的谱，尤其是六角系统图的谱。这些图的谱有助于估计对应的化合物的有关能量等其他方面的物理性质。

设图 G 是一个有 n 个顶点的图。图 G 的谱是指 G 的邻接矩阵的 n 个特征值构成的多重集合。有关图谱的研究是代数图论中的一个重要分支。图的谱与图结构有着密切联系。其中，化学图的谱在现实中存在广泛的应用背景。本项目主要研究某些化学图的谱，它可以反映相应的化合物的某些物理性质比如沸点、熔点等。我们得到如下主要结果：定义了一种新的图运算，并求出了经过这种图运算得到的图的谱；作为一种特殊情况，计算出了几类化学图的谱；利用组合数学中的递推公式计算出了Z型蜘蛛链的特征多项式，并得出了相应的化学指标；找到了一种可以构造无穷多类特征值互不相同的图的新方法；给出了杠铃图的谱的上界和下界。