Spectra of Random trees
نویسندگان
چکیده
We analyze the spectral distribution of the adjacency matrix and the graph Laplacian for a wide variety of random trees. Using soft arguments which seem to be applicable in a wide variety of settings, we show that the empirical spectral distribution for a number of random tree models, converges to a constant (model dependent) distribution. We also analyze the kernel of the spectrum and prove asymptotic convergence to limit constants for the kernel of the spectrum. We then go on to analyze the joint distribution of the maximal eigenvalues of the adjacency matrix in the linear preferential attachment model (with parameter a). We first show that the for any fixed k, the maximal k degrees rescaled properly converge in distribution. Using results of Frieze [20] and Chung et al [14], this implies that the k largest eigenvalues rescaled by na converge in distribution, where γa is the Malthusian rate of growth parameter for the associated continuous time branching process.
منابع مشابه
On the spectra of reduced distance matrix of the generalized Bethe trees
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