标题： 有实拉普拉斯谱的有向图 显示英文标题

标题： On Directed Graphs With Real Laplacian Spectra

摘要： 据报道，如果底层图的拉普拉斯矩阵具有纯实数谱，则有向图上的动力系统在系统阻尼和对时延的容忍方面表现出更优的性能。本文研究了有向图具有实数或复数拉普拉斯谱的拓扑条件。我们推导了可能包含自环和负权边的有向图具有实数拉普拉斯谱的充分条件。建立的条件通常意味着实数拉普拉斯谱与所谓的二边符号不对称相互作用以及有向图任何子图中非强连通性的缺失有关。然后，识别出两类具有复数拉普拉斯谱的有向图，这表明有向环的出现是导致复数拉普拉斯特征值的主要因素。此外，我们将分析扩展到多层有向图，提出了保持图互连中实数/复数谱的策略。数值实验表明，所得结果可以有效指导有向图拓扑结构的重新设计以获得更好的性能。 显示英文摘要

摘要： It is reported that dynamical systems over digraphs have superior performance in terms of system damping and tolerance to time delays if the underlying graph Laplacian has a purely real spectrum. This paper investigates the topological conditions under which digraphs possess real or complex Laplacian spectra. We derive sufficient conditions for digraphs, which possibly contain self-loops and negative-weighted edges, to have real Laplacian spectra. The established conditions generally imply that a real Laplacian spectrum is linked to the absence of the so-called digon sign-asymmetric interactions and non-strong connectivity in any subgraph of the digraph. Then, two classes of digraphs with complex Laplacian spectra are identified, which imply that the occurrence of directed cycles is a major factor to cause complex Laplacian eigenvalues. Moreover, we extend our analysis to multilayer digraphs, where strategies for preserving real/complex spectra from graph interconnection are proposed. Numerical experiments demonstrate that the obtained results can effectively guide the redesign of digraph topologies for a better performance.