### Some remarks on the square graph of the hypercube

#### Abstract

Let *Γ* = (*V*,*E*) be a graph. The *s**q**u**a**r**e* 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*_{n}^{2}. In particular, we show that the graph *Q*_{n}^{2} 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 *Q*_{n}^{2} 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}. Moreover, we show that when *n* > 2 is an even integer, then *Q*_{n}^{2} is an *a**u**t**o**m**o**r**p**h**i**c* graph, that is, *Q*_{n}^{2} is a distance-transitive primitive graph which is not a complete or line graph.

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MANUSCRIPTDOI: https://doi.org/10.26493/1855-3974.2621.26f

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications