Hermitian Laplacian Matrix of Directed Graphs

Abstract

Laplacian matrix plays an important role in the research of undirected graphs.From its spectrum,some structure and properties of a graph can be deduced.Based on this,several efficient algorithms have been designed for relevant tasks in graphs,such as graph partitioning and clustering.However,for directed graphs,the Laplacian is no longer symmetric,resulting in the complex eigenvalues,which are meaningless in most scenes.To circumvent this,the k'th root of unity are introduced as the weights of directed edges in recent researches,and the corresponding Hermitian Laplacian is defined.In this paper,the rotation angle of a directed edge and the generalized Hermitian Laplacian are introduced.It is shown that the Hermitian Laplacian has inherited some useful algebraic properties from ordinary Laplacian.To study the relations between directed graphs and the related Hermitian Laplacian,the definitions of constraint equations and directed cycle are proposed.It is proved that the following three statements are equivalent:1)the minimum eigenvalue of Hermitian Laplacian is equal to 0;2)the constraint equations have at least a solution;3)for each directed cycle in the graph,its rotation angle is equal to 2lπ($l \in \mathbb{Z}$).Finally,some corollaries and applications are presented.

Keywords

• spectral method|directed graph|hermitian laplacian matrix|directed cycle