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 $g>2d^{1/8}+1$, and if ${\lambda }_1\le {\lambda }_2\le \cdots {\lambda }_n$ are the eigenvalues of the (non-normalized) Laplacian of $G$, then \[ \frac {λ_n}{λ_2} \geq 1 + \frac 4{\sqrt d} - O \left( \frac 1{d^{\frac 58} }\right) \] (The Alon-Boppana theorem implies that if $G$ is unweighted and $d$-regular, then $\frac{{\lambda }_n}{{\lambda }_2}\ge 1+\frac{4}{\sqrt{d}}-O\left(\frac{1}{d}\right)$ if the diameter is at least $d^{1.5}$.) Our result implies a lower bound for spectral sparsifiers. A graph $H$ is a spectral $\epsilon$-sparsifier of a graph $G$ if \[ L(G) \preceq L(H) \preceq (1+ε) L(G) \] 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 $\epsilon$-sparsifier $H$ of average degree $d$ where $\epsilon \approx \frac{4\sqrt{2}}{\sqrt{d}}$ and the edges of $H$ are a (weighted) subset of the edges of $G$. Batson, Spielman and Srivastava also show that the bound on $\epsilon$ cannot be reduced below $\approx \frac{2}{\sqrt{d}}$ when $G$ is a clique; our Alon-Boppana-type result implies that $\epsilon$ cannot be reduced below $\approx \frac{4}{\sqrt{d}}$ 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 "online" setting. We show that for the online matrix setting the $4\sqrt{2}/\sqrt{d}$ bound is tight, up to lower order terms.