Cayley Properties of the Line Graphs Induced by 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 $$k+1$$ . The graph $$ {Q_n}(k,k+1) $$ has a close relation to Johnson graph $$J(n+1,k+1)$$ . In fact, it is the square root of the graph $$J(n+1,k+1)$$ . We will see that when $$ n\ne 2k+1$$ , then the graph $$ {Q_n}(k,k+1) $$ is a non-regular edge-transitive graph; hence, its line graph is a vertex-transitive graph. In the first step, we determine the automorphism groups of these graphs for all values of n, k. In the second step, we study Cayley properties of the line graphs of these graphs. In particular, we show that if $$k\ge 3$$ and $$ n \ne 2k+1$$ , then except for the cases $$(k,n) \ne (3,9)$$ and $$(k,n) \ne (3,33)$$ , 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. Moreover, we show that for ‘almost all’ even values of k, the line graph of the graph $$ {Q_{2k+1}}(k,k+1) $$ is a vertex-transitive non-Cayley graph.