The spectrum and automorphism group of the set-inclusion graph

Let $n$, $k$ and $l$ be integers with $1\le k<l\le n-1$. The set-inclusion graph $G(n,k,l)$ is the graph whose vertex set consists of all $k$- and $l$-subsets of $[n]=\{1,2,\dots ,n\}$, where two distinct vertices are adjacent if one of them is contained in another. In this paper, we determine the spectrum and automorphism group of $G(n,k,l)$, respectively.