Title: Cayley properties of the line graphs induced by of consecutive layers of the hypercube,
Abstract: Let $n >3$ and $ 0< k < \frac{n}{2} $ be integers. In this paper, we
investigate some algebraic properties of the line graph of the graph $
{Q_n}(k,k+1) $ where $ {Q_n}(k,k+1) $ is the subgraph of the hypercube $Q_n$
which is induced by the set of vertices of weights $k$ and $k+1$. In the first
step, we determine the automorphism groups of these graphs for all values of
$k$. In the second step, we study Cayley properties of the line graph of these
graphs. In particular, we show that for $ k>2, $ if $ 2k+1 \neq n$, then the
line graph of the graph $ {Q_n}(k,k+1) $ is a vertex-transitive non Cayley
graph. Also, we show that the line graph of the graph $ {Q_n}(1,2) $ is a
Cayley graph if and only if $ n$ is a power of a prime $p$.