For a positive integer $n > 1$, let $[n] = {1, 2, ..., n}$ and $V$ be the set of all k-subsets and $(n−k)$-subsets of $[n]$. The bipartite Kneser graph $H(n, k)$ has $V$ as its vertex set, and two vertices $A, B$ are adjacent if and only if $A ⊂ B$ or $B ⊂ A$.
We assume that $n ≥ 2k + 1$, or else the graph $H(n, k)$ would be null.
What is the connectivity of the bipartite Kneser graph $H(n, k)$? Is it just $^{n-k}C_n$? How can I approach this problem?
Further, is the graph $H(n, k)$ a symmetric (arc-transitive) graph?