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Consider the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices with fixed degree $d\geq 3$, and denote the eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We prove that the optimal (up to an extra $N^{{\rm o}_N(1)}$ factor, where ${\rm o}_N(1)$ can be arbitrarily small) eigenvalue rigidity holds. More precisely, denote $\gamma_i$ as the classical location of the $i$-th eigenvalue under the Kesten-Mckay law in decreasing order. Then with probability $1-N^{-1+{\rm o}_N(1)}$, \begin{align*} |\lambda_i-\gamma_i|\leq \frac{N^{{\rm o}_N(1)}}{N^{2/3} (\min\{i,N-i+1\})^{1/3}},\quad \text{ for all } i\in \{2,3,\cdots,N\}. \end{align*} In particular, the fluctuations of extreme eigenvalues are bounded by $N^{-2/3+{\rm o}_N(1)}$. This gives the same order of fluctuation as for the eigenvalues of matrices from the Gaussian Orthogonal Ensemble., Comment: 62 pages, 2 figures

We show that for an Erd\H{o}s-R\'{e}nyi graph on $N$ vertices with expected degree $d$ satisfying $\log^{-1/9}N\leq d\leq \log^{1/40}N$, the largest eigenvalues can be precisely determined by small neighborhoods around vertices of close to maximal degree. Moreover, under the added condition that $d\geq\log^{-1/15}N$, the corresponding eigenvectors are localized, in that the mass of the eigenvector decays exponentially away from the high degree vertex. This dependence on local neighborhoods implies that the edge eigenvalues converge to a Poisson point process. These theorems extend a result of Alt, Ducatez, and Knowles, who showed the same behavior for $d$ satisfying $(\log\log N)^4\ll d\leq (1-o_{N}(1))\frac{1}{\log 4-1}\log N$. To achieve high accuracy in the constant degree regime, instead of attempting to guess an approximate eigenvector of a local neighborhood, we analyze the true eigenvector of a local neighborhood, and show it must be localized and depend on local geometry., Comment: 52 pages

Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We show that for an arbitrary symmetric matrix, a positive fraction of eigenbases satisfy a generalized version of known nodal bounds for un-signed (that is classical) graphs. We do this through an explicit decomposition. Moreover, we show that with high probability, the number of nodal domains of a bulk eigenvector of the adjacency matrix of signed a Erd\H{o}s-R\'enyi graph is $\Omega(n/\log n)$ and $o(n)$., Comment: 39 pages 3 figure

We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets contained in both the left and right bipartitions exhibit unique-neighbor expansion, along with algebraic properties relevant to constructing quantum codes. Our constructions are obtained from instantiations of the tripartite line product of a large tripartite spectral expander and a sufficiently good constant-sized unique-neighbor expander, a new graph product we defined that generalizes the line product in the work of Alon and Capalbo and the routed product in the work of Asherov and Dinur. To analyze the vertex expansion of graphs arising from the tripartite line product, we develop a sharp characterization of subgraphs that can arise in bipartite spectral expanders, generalizing results of Kahale, which may be of independent interest. By picking appropriate graphs to apply our product to, we give a strongly explicit construction of an infinite family of $(d_1,d_2)$-biregular graphs $(G_n)_{n\ge 1}$ (for large enough $d_1$ and $d_2$) where all sets $S$ with fewer than a small constant fraction of vertices have $\Omega(d_1\cdot |S|)$ unique-neighbors (assuming $d_1 \leq d_2$). Additionally, we can also guarantee that subsets of vertices of size up to $\exp(\Omega(\sqrt{\log |V(G_n)|}))$ expand losslessly., Comment: New version contains stronger result, and many new technical ingredients. 45 pages, 2 figures

We prove quantum ergodicity for a family of periodic Schr\"odinger operators $H$ on periodic graphs. This means that most eigenfunctions of $H$ on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on $\mathbb{Z}^d$, the triangular lattice, the honeycomb lattice, Cartesian products and periodic Schr\"odinger operators on $\mathbb{Z}^d$. The theorem applies more generally to any periodic Schr\"odinger operator satisfying an assumption on the Floquet eigenvalues., Comment: Two important updates. (1) Wencai Liu arXiv:2210.10532 has solved the open problem of v1, so quantum ergodicity holds for periodic operators on $\mathbb{Z}^d$ in all dimensions. (2) We now prove the Floquet assumption cannot be dropped and replaced by mere ac spectrum. More additions and stronger conclusions are featured. 26 pages, 5 figures

Let $G$ be a random $d$-regular graph. We prove that for every constant $\alpha > 0$, with high probability every eigenvector of the adjacency matrix of $G$ with eigenvalue less than $-2\sqrt{d-2}-\alpha$ has $\Omega(n/$polylog$(n))$ nodal domains., Comment: 18 pages. Minor changes to the introduction

Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite., Comment: 12 pages, 3 figures

Kahale proved that linear sized sets in $d$-regular Ramanujan graphs have vertex expansion $\sim\frac{d}{2}$ and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than $\frac{d}{2}$. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether high-girth Ramanujan graphs have improved vertex expansion. Our results are two-fold: 1. For every $d = p+1$ for prime $p$ and infinitely many $n$, we exhibit an $n$-vertex $d$-regular graph with girth $\Omega(\log_{d-1} n)$ and vertex expansion of sublinear sized sets bounded by $\frac{d+1}{2}$ whose nontrivial eigenvalues are bounded in magnitude by $2\sqrt{d-1}+O\left(\frac{1}{\log n}\right)$. 2. In any Ramanujan graph with girth $C\log n$, all sets of size bounded by $n^{0.99C/4}$ have vertex expansion $(1-o_d(1))d$. The tools in analyzing our construction include the nonbacktracking operator of an infinite graph, the Ihara--Bass formula, a trace moment method inspired by Bordenave's proof of Friedman's theorem, and a method of Kahale to study dispersion of eigenvalues of perturbed graphs., Comment: 15 pages, 1 figure

We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree $\Delta$ is bounded by $O(n \Delta^{7/5}/\log^{1/5-o(1)}n)$ for any $\Delta$, and by $O(n\log^{1/2}d/\log^{1/4-o(1)}n)$ for simple $d$-regular graphs when $d\ge \log^{1/4}n$. In fact, the same bounds hold for the number of eigenvalues in any interval of width $\lambda_2/\log_\Delta^{1-o(1)}n$ containing the second eigenvalue $\lambda_2$. The main ingredient in the proof is a polynomial (in $k$) lower bound on the typical support of a closed random walk of length $2k$ in any connected graph, which in turn relies on new lower bounds for the entries of the Perron eigenvector of submatrices of the normalized adjacency matrix., Comment: A previous version of this paper proved the main result for d-regular graphs. The current version proves a more general result for the normalized adjacency matrix of bounded degree graphs. New version fixes an incorrect citation in the proof of Proposition 5.2. 24pp, 3 figures

In this paper, we study a general semi-random version of the planted independent set problem in a model initially proposed by Feige and Kilian, which has a large proportion of adversarial edges. We give a new deterministic algorithm that finds a list of independent sets, one of which, with high probability, is the planted one, provided that the planted set has size $k=\Omega(n^{2/3})$. This improves on Feige and Kilian's original randomized algorithm, which with high probability recovers an independent set of size at least $k$ when $k=\alpha n$ where $\alpha$ is a constant.