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We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary lengths. The implications of our results for the structural and dynamical properties of network models are discussed by showing how loops influence the size of the spectral gap and the propensity for synchronization. Analytical formulas for the spectrum are obtained for specific lengths of the loops.
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Metz et al. (2011) studied this question.
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