On the distance-transitivity of the folded hypercube

The folded hypercube F Qn is the Cayley graph Cay(Z n2, S), where S = {e1, e2, . . . , en} ∪ {u = e1 + e2 + · · · + en}, and ei = (0, . . . , 0, 1, 0, . . . , 0), with 1 at the ith position, 1 ≤ i ≤ n. In this paper, we show that the folded hypercube F Qn is a distance-transitive graph. Then, we study some properties of this graph. In particular, we show that if n ≥ 4 is an even integer, then the folded hypercube F Qn is an automorphic graph, that is, F Qn is a distance-transitive primitive graph which is not a complete or a line graph.