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 ≤ λ2 ≤ ⋯ λn are the eigenvalues of the (non-normalized) Laplacian of G, then [EQUATION] (The Alon-Boppana theorem implies that if G is unweighted and d-regular, then [EQUATION] (if the diameter is at least d1.5.) Our result implies a lower bound for spectral sparsifiers. A graph H is a spectral ϵ-sparsifier of a graph G if [EQUATION] 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 ϵ-sparsifier H of average degree d where [EQUATION] and the edges of H are a (weighted) subset of the edges of G. Batson, Spielman and Srivastava also show that the bound on ϵ cannot be reduced below [EQUATION] when G is a clique; our Alon-Boppana-type result implies that ϵ cannot be reduced below [EQUATION] when G comes from a family of expanders of super-constant degree and super-constant 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 setting. We show that for the online matrix setting the [EQUATION] bound is tight, up to lower order terms.