Pretty good state transfer on Cayley graphs over semi-dihedral groups
Dandan Wang, Xiwang Cao · Linear and Multilinear Algebra · 2021
Let Γ be a graph with adjacency matrix A. The transition matrix of Γ corresponding to A is defined by H(t):=exp(−ιtA), where ι=−1 and t∈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 {tk} and a complex number γ with unit norm such that limk→∞H(tk)eu=γev. 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.