Some remarks on the square graph of the hypercube

Let $Γ=(V,E)$ be a graph. The square graph $Γ^2$ of the graph $Γ$ is the graph with the vertex set $V(Γ^2)=V$ in which two vertices are adjacent if and only if their distance in $Γ$ 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.

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