s-距离传递（有向）图的有限性猜想研究

Distance transitive (di-)graphs have highly symmetry, and are the important research objects in the field of groups and graphs. In 1983, Cameron et al. proved the finiteness conjecture of distance-transitive graphs: the number of distance-transitive graphs with given valency greater than two is finite. In 1993, Leonard and Nomura proved the finiteness conjecture of distance-transitive digraphs. Inspired by the proofs of above finiteness conjectures, that to give characterization and classification of general s-distance-transitive (di-)graphs, and then to give the proof of the finiteness conjecture, are the important problems considered by mathematicians in and abroad. Near-distance-transitive (di-)graphs is a special class of s-distance-transitive (di-)graphs such that s is equal the diameter of the (di-)graph minus one. This project is going to give the proof for the finiteness conjecture of near-distance-transitive (di-)graphs. By studying the characterizations and properties of distance-transitive (di-)graphs with small valency, basic 2-arc-transitive near-distance-transitive (di-)graphs and their covers, independent near-distance-transitive graphs, and distance matrices of near-distance-transitive (di-)graphs, we will then have deep research on near-distance-transitive (di-)graphs, and finally we will give the proof of the finiteness of near-distance-transitive (di-)graphs, which is the first and most important thing for the proof of the finiteness conjecture of general s-distance-transitive (di-)graphs.

距离传递（有向）图具有高度的对称性，是群与图领域非常重要的研究对象。1983年，Cameron等人证明了距离传递图的有限性猜想：给定度数大于2的距离传递图的个数有限。1993年，Leonard和Nomura证明了距离传递有向图的有限性猜想。受上述有限性猜想被证明的启发，对一般的s-距离传递（有向）图的性质研究和刻画分类, 并由此给出其有限性猜想的证明成为国内外数学家关注的重要问题。接近距离传递（有向）图是一类特殊的s-距离传递（有向）图，它满足s等于图的直径减一。本项目拟通过对小度数接近距离传递（有向）图、基本的2-弧传递的接近距离传递（有向）图及其覆盖、独立的接近距离传递图、以及接近距离传递（有向）图的距离矩阵的刻画和性质研究，对接近距离传递（有向）图进行深入研究，从而给出接近距离传递（有向）图有限性证明，为解决一般的s-距离传递（有向）图的有限性猜想提供理论方法和工具。