标题： 关于两个距离传递图 显示英文标题

标题： On two-distance-transitive graphs

摘要： 一个$2$-距离传递图是一个顶点传递图，其顶点稳定子在第一邻域和第二邻域上都是传递的。 在本文中，我们首先回答了A. Devillers, M. Giudici, C. H. Li和C. E. Praeger在2012年关于奇数阶的$2$-距离传递图的问题。 然后我们对度数为$p$或$p+1$的$2$-距离传递图进行了刻画，其中$p$是一个素数。 之后，作为上述结果的应用，我们对小度数的局部本原、$2$-距离传递图进行了分类。 除了上述关于$2$-距离传递图的结果外，我们还对直径至少为$4$且参数为$(v, k, \lambda, \frac{k - 1}{2})$的强正则图族进行了分类，这些图在局部本原、$2$-距离传递图且小度数的分类中自然出现。 显示英文摘要

摘要： A $2$-distance-transitive graph is a vertex-transitive graph whose vertex stabilizer is transitive on both the first step and the second step neighborhoods. In this paper, we first answer a question of A. Devillers, M. Giudici, C. H. Li and C. E. Praeger in 2012 about vertex-quasiprimitive $2$-distance-transitive graphs for the odd order case. Then we characterize $2$-distance-transitive graphs of valency $p$ or $p+1$ where $p$ is a prime. After that, as an application of the above result, we classify locally-primitive, $2$-distance-transitive graphs of small valency. In addition to the above results on $2$-distance-transitive graphs, we also classify a family of amply regular graphs with diameter at least $4$ and parameters $(v, k, \lambda, \frac{k - 1}{2})$, and these graphs arise naturally in the classification of locally-primitive, $2$-distance-transitive graphs with small valency.