Line graphs of complex unit gain graphs with least eigenvalue -2

Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph $\mathbb{T}$-gain $\Phi$. Similarly to what happens in context signed graphs, real number $\min Spec(A(\mathcal (\Phi))$, that is, smallest eigenvalue adjacency matrix L(\Phi)$, is not less than $-2$. The structural conditions on $\Phi$ ensuring (\Phi))=-2$ are identified. When such fulfilled, bases $-2$-eigenspace constructed with aid star complement technique.