The automorphism group of the bipartite Kneser graph

Let $n$ and $k$ be integers with $n>2k,k\ge 1$. We denote by $H(n,k)$ the $bipartiteKnesergraph$, that is, a graph with the family of $k$-subsets and ($n-k$)-subsets of $[n]=\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ as vertices, in which any two vertices are adjacent if and only if one of them is a subset of the other. In this paper, we determine the automorphism group of $H(n,k)$. We show that $Aut(H(n,k))\cong Sym([n])\times {\mathbb{Z}}_2$ where ${\mathbb{Z}}_2$ is the cyclic group of order $2$. Then, as an application of the obtained result, we give a new proof for determining the automorphism group of the Kneser graph $K(n,k)$. In fact we show how to determine the automorphism group of the Kneser graph $K(n,k)$ given the automorphism group of the Johnson graph $J(n,k)$. Note that the known proofs for determining the automorphism groups of Johnson graph $J(n,k)$ and Kneser graph $K(n,k)$ are independent from each other.