Spectra of sparse regular graphs with loops

Abstract

We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks 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 length of the loops.

4 pages, 4 figures

References in corpus (11)

• Synchronization is optimal in non-diagonalizable networks
• Cavity Approach to the Spectral Density of Sparse Symmetric Random Matrices
• Spectra of Sparse Random Matrices
• Laplacian spectra of complex networks and random walks on them: Are scale-free architectures really important?
• Local structure of directed networks
• Cavity approach to the spectral density of non-Hermitian sparse matrices
• On the localization transition in symmetric random matrices
• Anderson model on Bethe lattices: density of states, localization properties and isolated eigenvalue
• Hopping on the Bethe lattice: Exact results for densities of states and dynamical mean-field theory
• Spectral properties of the Google matrix of the World Wide Web and other directed networks
• Spectra of Modular and Small-World Matrices

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