Note on the spectra of a class of graphs derived from set inclusion relations.

For any given integers $n$, $k$ and $l$ with $n\geq 1$ and $0\leq k<l\leq n$, we denote by $G(n,k,l)$ the graph whose vertex set consists of all $k$- and $l$-subsets of $[n]=\{1,2,\ldots,n\}$, where two distinct vertices are adjacent if one of them is contained in another. In this note, we determine the spectrum of $G(n,k,l)$ (and its line graph) for arbitrary $k,l$. As by-products, we obtain the spectra of the subgraphs (and their line graphs) of the hypercube induced by two consecutive layers, which generalizes a result due to Mirafzal [S.M. Mirafzal, A new class of integral graphs constructed from the hypercube, Linear Algebra Appl. 558 (2018) 186--194].