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

Abstract

Let G be a complex unit gain graph which is obtained from an undirected graph Γ by assigning a complex unit φ(vivj) to each oriented edge vivj such that φ(vivj)φ(vjvi)=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 Γ and A(G)=(aij) has aij=φ(vivj) if vi is adjacent to vj and aij=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.