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 = vi1vi2 vikvi1 we have ϕ(vi1vi2)ϕ(vi2vi3) ϕ(vikvi1) = 1. It is known that Φ is balanced if and only if the least Laplacian eigenvalue λn(Φ) is 0. Here we show that, if Φ is unbalanced and ϕ(Φ) ⊆ 4, the eigenvalue λn(Φ) measures how far is Φ from being balanced. More precisely, let ν(Φ) (respectively, ∈(Φ)) be the number of vertices (respectively, edges) to cancel in order to get a balanced gain subgraph. We show that λn(Φ) ≤ ν(Φ) ≤ ∈(Φ). We also analyze the case when λn(Φ) = ν(Φ). In fact, we identify the structural conditions on Φ that lead to such equality.