Some remarks on the square graph of the hypercube

Let Γ = (V,E) be a graph. The square graph Γ2 of the graph Γ is the graph with the vertex set V(Γ2) = 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 Qn has some interesting properties. For instance, it is highly symmetric and panconnected.In this paper, we investigate some algebraic properties of the graph Qn2. In particular, we show that the graph Qn2 is distance-transitive. We will see that this property, in some aspects, is an outstanding property in the class of distance-transitive graphs. We show that the graph Qn2 is an imprimitive distance-transitive graph if and only if n is an odd integer. Also, we determine the spectrum of the graph Qn2. Moreover, we show that when n > 2 is an even integer, then Qn2 is an automorphic graph, that is, Qn2 is a distance-transitive primitive graph which is not a complete or line graph.