On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s$-th (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon--Boppana type bound for a class of irregular graphs.Comment: 12 page

An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification

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 $\lambda_1 \leq \lambda_2 \leq \cdots \lambda_n$ are the eigenvalues of the (non-normalized) Laplacian of $G$, then $$\frac {\lambda_n}{\lambda_2} \geq 1 + \frac 4{\sqrt d} - O \left( \frac 1{d^{\frac 58} }\right)$$ (The Alon-Boppana theorem implies that if $G$ is unweighted and $d$-regular, then $\frac {\lambda_n}{\lambda_2} \geq 1 + \frac 4{\sqrt d} - O\left( \frac 1 d \right)$ 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 $$L(G) \preceq L(H) \preceq (1+\epsilon) L(G)$$ 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

Spectral radii of sparse random matrices

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erd\H{o}s-R\'enyi graphs. Our error bounds are sharp for a large class of sparse random matrices. In particular, for the Erd\H{o}s-R\'enyi graph $G(n,d/n)$, our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that $d \gg \log n$. Together with the companion paper [3], where we analyse the extreme eigenvalues in the complementary regime $d \ll \log n$, this establishes a crossover in the behaviour of the extreme eigenvalues around $d \sim \log n$. Our results also apply to non-Hermitian sparse random matrices, corresponding to adjacency matrices of directed graphs. The proof combines (i) a new inequality between the spectral radius of a matrix and the spectral radius of its nonbacktracking version together with (ii) a new application of the method of moments for nonbacktracking matrices

Spectra of lifted Ramanujan graphs

A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices $[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent uniform bijection $\pi$, and $H$ has all edges of the form $(i,u),(\pi(i),v)$. A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan. Let $G$ be a graph with largest eigenvalue $\lambda_1$ and let $\rho$ be the spectral radius of its universal cover. Friedman (2003) proved that every "new" eigenvalue of a random lift of $G$ is $O(\rho^{1/2}\lambda_1^{1/2})$ with high probability, and conjectured a bound of $\rho+o(1)$, which would be tight by results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved Friedman's bound to $O(\rho^{2/3}\lambda_1^{1/3})$. For $d$-regular graphs, where $\lambda_1=d$ and $\rho=2\sqrt{d-1}$, this translates to a bound of $O(d^{2/3})$, compared to the conjectured $2\sqrt{d-1}$. Here we analyze the spectrum of a random $n$-lift of a $d$-regular graph whose nontrivial eigenvalues are all at most $\lambda$ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is $O((\lambda \vee \rho) \log \rho)$. This result is tight up to a logarithmic factor, and for $\lambda \leq d^{2/3-\epsilon}$ it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical $n$-lift of a Ramanujan graph is nearly Ramanujan.Comment: 34 pages, 4 figure

Cycle density in infinite Ramanujan graphs

We introduce a technique using nonbacktracking random walk for estimating the spectral radius of simple random walk. This technique relates the density of nontrivial cycles in simple random walk to that in nonbacktracking random walk. We apply this to infinite Ramanujan graphs, which are regular graphs whose spectral radius equals that of the tree of the same degree. Kesten showed that the only infinite Ramanujan graphs that are Cayley graphs are trees. This result was extended to unimodular random rooted regular graphs by Ab\'{e}rt, Glasner and Vir\'{a}g. We show that an analogous result holds for all regular graphs: the frequency of times spent by simple random walk in a nontrivial cycle is a.s. 0 on every infinite Ramanujan graph. We also give quantitative versions of that result, which we apply to answer another question of Ab\'{e}rt, Glasner and Vir\'{a}g, showing that on an infinite Ramanujan graph, the probability that simple random walk encounters a short cycle tends to 0 a.s. as the time tends to infinity.Comment: Published at http://dx.doi.org/10.1214/14-AOP961 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org