Some remarks on the square graph of the hypercube - Mirafzal, Seyed Morteza (Ars mathematica contemporanea, 2023)
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. An interesting property of the square graph of the hypercube ?$Q_n$? is that it is highly symmetric and panconnected. In this article, the author investigates some algebraic properties of the graph ?$Q^2_n$?. He shows that the graph ?$Q^2_n$? is distance-transitive and that the graph ?$Q^2_n$? is an imprimitive distance-transitive graph if and only if ?$n$? is an odd integer and determines the spectrum of the graph ?$Q^2_n$?. Finally, the author shows that when ?$n > 2$? is an even integer, then ?$Q^2_n$? is an automorphic graph, that is, ?$Q^2_n$? is a distance-transitive primitive graph which is not a complete or a line graph.
Keywords: automorphic graph; automorphism group; avtomorfni graf; distance-transitive graph; grupa avtomorfizmov; hiperkocka; hypercube; Johnson graph; Johnsonov graf; kvadrat grafa; razdaljno tranzitiven graf; square of a graph