Matrix-Tree Theorems for complex unit gain networks

Let T be the group of all complex numbers z with norm 1. The celebrated Kirchhoff's Matrix-Tree Theorem on electrical networks has been generalized in many directions. In this paper we consider its version for the complex unit gain networks, i.e. pairs of type Φ=(Γ,φ), where Γ=(VΓ,EΓ) is a network and φ is a map from the arcs E→Γ of Γ to T such that φ(uv)=φ(vu)−1 for every arc uv. A complex unit gain variant of Kelmans's Matrix-Tree Theorem will be also discussed. Of both Kirchhoff's and Kelmans's Theorems we provide elementary proofs only relying on the factorization of Laplacian matrix into an incidence matrix and its Hermitian transpose. As a consequence, we retrieve the Matrix-Tree Theorems for signed and mixed networks.