距离正则图最小特征值的若干问题

Distance-regular graph is an important topic in algebraic combinatorics. The smallest eigenvalue is of general interest in the study of distance-regular graphs and it has close relations with many properties of distance-regular graphs, like the chromatic number, girth and geometric distance-regular graphs. This project concerns the smallest eigenvalue problems of distance-regular graphs from three aspects: (1) Some problems on the smallest eigenvalue about the classification of 3-chromatic distance-regular graphs. (2)The classifications of some classes of geometric distance-regular graphs with good structures, and the characterizations of some classes of distance-regular graphs using geometric distance-regular graphs. (3) The relations between the girth and lower bound of the smallest eigenvalue of distance-regular graphs, and the tight lower bound of the smallest eigenvalue when the girth is small.

距离正则图是代数组合的重要课题，最小特征值是距离正则图研究的热点问题，它与距离正则图的色数和围长、几何距离正则图等性质都有着密切的联系。本项目主要从三个方面研究距离正则图的最小特征值问题：（1）3色距离正则图分类相关的最小特征值问题。（2）一些结构良好的几何距离正则图类的分类问题和利用几何距离正则图对一些距离正则图类的刻画问题。（3）距离正则图的围长与最小特征值下界的关系和围长较小时最小特征值的严格下界。

距离正则图是代数组合的重要课题，最小特征值是距离正则图研究的热点问题，它与距离正则图的色数和围长、几何距离正则图等性质都有着密切的联系。在对距离正则图最小特征值相关问题的研究中，本项目主要取得以下结果：1)利用几何距离正则图的性质，给出了对偶极图的一种新的刻画，从而进一步分类了3色距离正则图；2)研究了非二部距离正则图的最小特征值与奇围长的关系，进一步分类了最小特征值接近-k的非二部距离正则图；3)研究了距离正则图Cheeger常数的相关猜想，证明了该猜想对已知的距离正则图无穷类成立。