{"ID":2885087,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.05150","arxiv_id":"2508.05150","title":"On Directed Graphs With Real Laplacian Spectra","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2508.05150","url_pdf":"https://arxiv.org/pdf/2508.05150v1","authors":"[\"Tianhao Yu\",\"Shenglu Wang\",\"Mengqi Xue\",\"Yue Song\",\"David J. Hill\"]","published":"2025-08-07T08:35:38Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}