二部Kneser图的一些代数性质
Some algebraic properties of bipartite Kneser graphs
摘要：
令$n$和$k$是整数，其中$n> k\geq1$和$[n] = \{1, 2, ... , n\} $。$bipartite \ Kneser \ graph$ $H(n, k)$是以所有$k$ -元素和$[n] $的所有（$n-k$）-元素子集作为顶点的图，并且当一个是另一个的子集时，任何两个顶点之间存在边。本文证明了$H(n, k)$是弧传递图。同时，我们证明了$H(n,1)$是一个距离传递的Cayley图.最后，我们确定了图$H(n, 1)$的自同构群，并证明了$Aut(H(n, 1)) \cong Sym([n] )$ $\times \mathbb{Z}_2$，其中$\mathbb{Z}_2$是阶为$2$的循环群.此外，我们还提出了关于二部Kneser图$H(n, k)$的自同构群的一些公开问题。
英文摘要：
Let $n$ and $k$ be integers with $n> k\geq1$ and $[n] = \{1, 2, ... , n\} $. The $bipartite \ Kneser \ graph$ $H(n, k)$ is the graph with the all $k$-element and all ($n-k$)-element subsets of $[n] $ as vertices, and there is an edge between any two vertices, when one is a subset of the other. In this paper, we show that $H(n, k)$ is an arc-transitive graph. Also, we show that $H(n,1)$ is a distance-transitive Cayley graph. Finally, we determine the automorphism group of the graph $H(n, 1)$ and show that $Aut(H(n, 1)) \cong Sym([n] )$ $\times \mathbb{Z}_2$, where $\mathbb{Z}_2$ is the cyclic group of order $2$. Moreover, we pose some open problems about the automorphism group of the bipartite Kneser graph $H(n, k)$.
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