The spectrum of random k-lifts of large graphs (with possibly large k)

We study random k-lifts of large, but otherwise arbitrary graphs G. We prove that, with high probability, all eigenvalues of the adjacency matrix of the lift that are not eigenvalues of G are of the order of √ ΔG ln(kn), where ΔG is the maximum degree of G. Similarly, and also with high probability, the “new” eigenvalues of the normalized Laplacian of G are all in an interval of length O( √ ln(nk)/δG) around 1, where δG is the minimum degree of G. We also prove that, from the point of view of spectra, there is very little difference between a random k1 k2 . . . kr-lift of a graph and a random k1-lift of a random k2-lift of . . . of a random kr-lift of the same graph. The main proof tool is a concentration inequality for sums of independent random matrices which is of independent interest.