An Alon-Boppana type bound for weighted graphs and lowerbounds for spectral sparsification

Abstract

We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if G is an n-node weighted undirected graph of average combinatorial degree d (that is, G has dn=2 edges) and girth g > 2d1=8+1, and if λ1 ≥ λ2λ n are the eigenvalues of the (nonnormalized) Laplacian of G, then λn/λ2 ≥ 1 + 4/√ d o(1/d5/8) (The Alon-Boppana theorem implies that if G is unweighted and d-regular, then λn/λ2 ≥ 1 + √p/4/d - O -1 d if the diameter is at least d1:5.) Our result implies a lower bound for spectral sparsifiers. A graph H is a spectral-sparsifier of a graph G if L(G) ≺ L(H) ≺ (1 +ϵ)L(G) where L(G) is the Laplacian matrix of G and L(H) is the Laplacian matrix of H. Batson, Spielman and Srivastava proved that for every G there is an -sparsifier H of average degree d where ϵ 4 √2/√d and the edges of H are a (weighted) subset of the edges of G. Batson, Spielman and Srivastava also show that the bound on epsi; cannot be reduced below epsi; p2 d when G is a clique; our Alon-Boppana-type result implies that epsi; cannot be reduced below epsi; p4 d when G comes from a family of expanders of super-constant degree and superconstant girth. The method of Batson, Spielman and Srivastava proves a more general result, about sparsifying sums of rank-one matrices, and their method applies to an "online" setting. We show that for the online matrix setting the 4√2/√d bound is tight, up to lower order terms.