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

TL;DR

This paper analyzes the spectral properties of random k-lifts of large graphs, showing that new eigenvalues are tightly bounded and that multi-lift constructions behave similarly to single-lift cases.

Contribution

It provides probabilistic bounds on eigenvalues of adjacency and Laplacian matrices for random k-lifts, introducing new spectral concentration results for these graph transformations.

Findings

New eigenvalues are bounded by (D ln (kn))^{1/2} with high probability.

Lifts of different sizes have similar spectral properties.

Spectral behavior of multi-lifts approximates that of single-lifts.

Abstract

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 (D ln (kn))^{1/2}, where D is the maximum degree of G. Similarly, and also with high probability, the "new" eigenvalues of the Laplacian of the lift are all in an interval of length (ln (nk)/d)^{1/2} around 1, where d is the minimum degree of G. We also prove that, from the point of view of Spectral Graph Theory, there is very little difference between a random k_1k_2 ... k_r-lift of a graph and a random k_1-lift of a random k_2-lift of ... of a random k_r-lift of the same graph. The main proof tool is a concentration inequality for sums of random matrices that was recently introduced by the author.