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We study the symmetry properties of the spectra of normalized Laplacians onsigned graphs. We find a new machinery that generates symmetric spectra forsigned graphs, which includes bipartiteness of unsigned graphs as a specialcase. Moreover, we prove a fundamental connection between the symmetry of thespectrum and the existence of damped two-periodic solutions for thediscrete-time heat equation on the graph.

On the symmetry of the Laplacian spectra of signed graphs