Spectra of sparse regular graphs with loops

Details Spectra of sparse regular graphs with loops Spectra of sparse regular graphs with loops

2011-07-20

arxiv.org/abs/1107.4127v1

Phys. Rev. E 84, 055101(R) (2011)

Physics

Condensed Matter

Statistical Mechanics

4 pages, 4 figures

Scientific paper

10.1103/PhysRevE.84.055101

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.

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