Orthogonal and unitary signings of cube-like graphs

A unitary signing of a $d$-regular graph is a Hermitian adjacency matrix $M$ whose nonzero entries lie in $\{\pm1,\pm i\}$ and satisfies $M^2=dI$. Motivated by the work of Alon and Zheng on orthogonal and unitary signings of cube-like graphs, we introduce the $Θ$-property for a generating set $S\sub