Line and subdivision graphs determined by T4-gain graphs

Let T4 = (±1, ±i) be the subgroup of fourth roots of unity inside T, the multiplicative group of complex units. For a T4-gain graph Φ = (Γ,T4, ϕ), we introduce gain functions on its line graph L(Γ) and on its subdivision graph S(Γ). The corresponding gain graphs L(Φ) and S(Φ) are defined up to switching equivalence and generalize the analogous constructions for signed graphs. We discuss some spectral properties of these graphs and in particular we establish the relationship between the Laplacian characteristic polynomial of a gain graph Φ, and the adjacency characteristic polynomials of L(Φ) and S(Φ). A suitably defined incidence matrix for T4-gain graphs plays an important role in this context