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 ( v i v j ) to each oriented edge v i v j such that φ ( v i v j ) φ ( v j v i ) = 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 ) = ( a i j ) has a i j = φ ( v i v j ) if v i is adjacent to v j and a i j = 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.

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