The degree sequences and spectra of scale‐free random graphs

We investigate the degree sequences of scale-free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non-rigorous arguments of Dorogovtsev et al [10]. We also consider a generalisation of the model with more randomisation, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localised eigenfunctions of the adjacency matrix can be found. 2

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