Pretty good state transfer on Cayley graphs over semi-dihedral groups

Let Gamma be a graph with adjacency matrix A. The transition matrix of Gamma corresponding to A is defined by H(t) := exp(-iota tA), where iota = root-1 and t is an element of R. The graph is said to exhibit pretty good state transfer between a pair of vertices u and v if there exists a sequence of real numbers {t(k)} and a complex number gamma with unit norm such that lim(k ->infinity) H(t(k))e(u) = gamma e(v). In this paper, we explore pretty good state transfer on Cayley graphs over semi-dihedral groups by using the representations of such groups. We show that graphs Cay(SD8n, S) have pretty good state transfer for some suitable subsets S if n is a power of 2. Moreover, we present a sufficient and necessary condition for a non-integral graph Cay(SD8n, S) to admit pretty good state transfer. Some concrete constructions of Cayley graphs over semi-dihedral groups having PGST are also presented.