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.

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 $Ω(\log_{d-1} n)$ 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.

We show that the multiplicity of the second normalized adjacency matrix eigenvalue of any connected graph of maximum degree Δ is bounded by O(n Δ7/5/log1/5−o(1)n) for any Δ, and improve this to O(nlog1/2d/log1/4−o(1)n) for simple d-regular graphs when d≥ log1/4n. In fact, the same bounds hold for the number of eigenvalues in any interval of width λ2/logΔ1−o(1)n containing the second eigenvalue λ2. 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.

We study the independent set problem in a semi-random model proposed by Feige and Kilian. This model selects a graph with a planted independent set of size k and then allows an adversary to modify a large fraction of edges: the subgraph induced by the complement of the independent set can be modified arbitrarily, and the adversary may add (but not delete) edges from the independent set to its complement. In particular, the adversary can create a graph in which the initial planted independent set is not the largest independent set. Feige and Kilian presented a randomized algorithm, which with high probability recovers an independent set of size at least k (which may not be the planted one) when k = an where a is a constant, and the probability of a random edge p > (1 + ϵ) ln n/αn. We give a new deterministic algorithm in the Feige-Kilian model that finds an independent set of size at least .99k provided that the planted set has size k = Ω(n2/3/p1/3), and finds a list of independent sets, one of which is the planted one provided that k = Ω(n2/3/p). This improves on the algorithm of Feige and Kilian by working for smaller k if p = Ω(1/n1/3), and improves on an algorithm of Steinhardt by working for slightly smaller k and by working against a stronger adversarial model. The ability to find a good approximation of the largest independent set is new when p < ln n/k.