Perfect State Transfer on Oriented Circulant Graphs: A Complete Classification

Abstract

The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor $\Delta$, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order $n$, we derive an explicit formula for every Fourier eigenvalue without assuming that $n/\Delta$ is coprime to $\Delta$. We prove that PST occurs only for $\Delta\in\{3,4,8\}$ and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly $1$, $2$, and $3$. More generally, when $\lambda_j=\sqrt{D}\eta_j$ with $\eta_j\in\mathbb{Z}$, congruences satisfied by the integers $\eta_j$ determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.