Cayley properties of the line graphs induced by consecutive layers of the hypercube

TL;DR

This paper explores the algebraic and Cayley properties of line graphs derived from specific subgraphs of hypercubes, revealing conditions under which these graphs are vertex-transitive or Cayley, with some exceptions.

Contribution

It determines automorphism groups of these line graphs and characterizes when they are Cayley or vertex-transitive, including new results for various parameter ranges.

Findings

Line graphs are mostly vertex-transitive non-Cayley graphs.

Line graph of ${Q_n}(1,2)$ is Cayley iff n is a prime power.

Almost all line graphs of ${Q_{2k+1}}(k,k+1)$ are vertex-transitive non-Cayley graphs.

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 $n,k$. In the second step, we study Cayley properties of the line graphs of these graphs. In particular, we show that if $k\geq 3$ and $ n \neq 2k+1$, then except for the cases $k=3, n=9$ and $k=3, n=33$, 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 \lq{}almost all\rq{} even values of $k$, the line graph of the graph $ {Q_{2k+1}}(k,k+1) $ is a vertex-transitive non-Cayley graph.