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

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