Perfect state transfer on mixed graphs: complete classes and transfer times

Abstract

For perfect state transfer (PST) on unweighted mixed graphs, we classify the normalized transfer times of complete PST classes. A finite set $Λ\subset\mathbb R/\mathbb Z$ containing zero occurs at a nonstationary periodic vertex if and only if $\cos(2π(x-y))\in\mathbb Q$ for all $x,y\inΛ$. Every admissible set has a connected oriented realization. We also classify the possible return phases of oriented realizations at the minimum vertex period. Transfers at rational multiples of the common minimum vertex period partition a complete class into sets of size at most six, or at most three in an oriented graph with return phase $-1$; both bounds are sharp. We construct complete classes of every finite size, including classes in which all transfers between distinct vertices occur at irrational multiples of the period and no switching automorphism maps a class vertex to a distinct class vertex. We also characterize simultaneous realization in connected oriented graphs with prescribed relative minimum vertex periods and return phases. The proof combines an imaginary quadratic field restriction with an unweighted construction that selects the complete target set.