Some remarks on the square graph of the hypercube

TL;DR

This paper explores the algebraic and symmetry properties of the square graph of hypercubes, revealing its distance-transitivity, spectrum, and automorphic characteristics depending on the dimension's parity.

Contribution

It establishes that the square of the hypercube is distance-transitive, characterizes its spectrum, and identifies conditions for its automorphic and primitive properties.

Findings

${Q^2_n}$ is distance-transitive.

${Q^2_n}$ is imprimitive if and only if $n$ is odd.

For even $n > 2$, ${Q^2_n}$ is an automorphic, primitive, non-complete, non-line graph.

Abstract

Let $\Gamma=(V,E)$ be a graph. The square graph $\Gamma^2$ of the graph $\Gamma$ is the graph with the vertex set $V(\Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $\Gamma$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n >2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.