On the determinant of the Laplacian matrix of a complex unit gain graph

Let G be a complex unit gain graph which is obtained from an undirected graph Gamma by assigning a complex unit phi(upsilon(i)upsilon(j) ) to each oriented edge upsilon(i)upsilon(j) such that phi(upsilon(i),upsilon(j))phi(upsilon(j)upsilon(1)) = 1 for all edges. The Laplacian matrix of G is defined as L(G) = D(G) A(G), where D(G) is the degree diagonal matrix of Gamma and A(G) = (a,(j)) has a(ij) = phi(upsilon(i)upsilon(j) ) if vi is adjacent to upsilon(j) and a(ij) = 0 otherwise. In this paper, we provide a combinatorial description of det(L(G)) that generalizes that for the determinant of the Laplacian matrix of a signed graph. (C) 2017 Elsevier B.V. All rights reserved.