Balancedness and the Least Laplacian Eigenvalue of Some Complex Unit Gain Graphs

Let 4 = {±1, ±i} be the subgroup of 4-th roots of unity inside , the multiplicative group of complex units. A complex unit gain graph Φ is a simple graph Γ = (V (Γ) = {v1, . . . , vn}, E(Γ)) equipped with a map ϕ:E→(Γ)→\varphi :\vec E(\Gamma ) \to \mathbb{T} defined on the set of oriented edges such that ϕ(vivj) = ϕ(vjvi)−1. The gain graph Φ is said to be balanced if for every cycle C = vi1vi2vikvi1 we have ϕ(vi1vi2)ϕ(vi2vi3) ϕ(vikvi1) = 1