  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 $H(n, k)$.
