The automorphism group of the bipartite Kneser graph

Abstract

Let n and k be integers with $n>2k$, $k\ge 1$. We denote by H(n, k) the bipartite Kneser graph, that is, a graph with the family of k-subsets and ($n-k$)-subsets of $[n] = \{1, 2,\ldots , n\}$ 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 $\mathrm{Aut}(H(n, k))\cong \mathrm{Sym}([n]) \times {\mathbb {Z}}_2$, 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 of each other.