距离正则图是一个重要的图类, 它等价于对称 P-多项式结合方案, 与图论、有限几何、设计、编码等有着密切联系. 作为距离正则图的有向情形, 王恺顺和 Suzuki 在 2003 年提出了弱距离正则有向图, 这类有向图对应于非对称结合方案. 本文首次研究具有 P-多项式性质的弱距离正则有向图, 刻画了对应结合方案是 P-多项式的情形, 给出了基图分别为 4 类经典距离正则图的交换弱距离正则有向图的完全分类.

Distance-regular graphs are an important class of graphs, which are equivalent to symmetric P-polynomial association schemes and have close connections to graph theory, finite geometry, coding theory, and design theory. As a directed version of distance-regular graphs, weakly distance-regular digraphs were proposed by Wang and Suzuki in 2003, which correspond to non-symmetric association schemes. In this thesis, we initiate studying weakly distance-regular digraphs with P-polynomial property, characterize weakly distance-regular digraphs whose attached schemes are $P$-polynomial, and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded $n$-cubes, Doob graphs, and Johnson graphs, respectively.