Distance powers of integral Cayley graphs over dihedral groups and dicyclic groups

Abstract

In this paper, we focus on the dihedral groups and the dicyclic groups, and consider their corresponding integral Cayley graphs. We obtain the sufficient conditions for the integrality of the distance powers ᴦD of the Cayley graph ᴦ = X(D2n, S) (resp. ᴦ = X(D4n, S)) (n ≥ 3)) for a set of nonnegative integers D. In particular, for a prime p, we show that if ᴦ = X(D2p, S) (resp. ᴦ = X(D4p, S)) is integral, then the distance powers of ᴦ = X(D2p, S) (resp. ᴦ = X(D4p, S)) are integral Cayley graphs.