New Explicit Constant-Degree Lossless Expanders

We present a new explicit construction of onesided bipartite lossless expanders of constant degree, with arbitrary constant ratio between the sizes of the two vertex sets. Our construction is simpler to state and analyze than the only prior construction of Capalbo, Reingold, Vadhan, and Wigderson (2002), and achieves improvements in some parameters. We construct our lossless expanders by imposing the structure of a constant-sized lossless expander "gadget" within the neighborhoods of a large bipartite spectral expander; similar constructions were previously used to obtain the weaker notion of unique-neighbor expansion. Our analysis simply consists of elementary counting arguments and an application of the expander mixing lemma.Comment: Edits to expositio

Explicit near-Ramanujan graphs of every degree

For every constant $d \geq 3$ and $\epsilon > 0$, we give a deterministic $\mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$ vertices that is $\epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2\sqrt{d-1} + \epsilon$ (excluding the single trivial eigenvalue of~$d$).Comment: 26 page

A Combinatorial Construction of Almost-Ramanujan Graphs Using the Zig-Zag Product

introduced the graph zig-zag product. This product combines a large and a small graph into one, such that the resulting graph inherits its size from the large graph, its degree from the small graph, and its spectral gap from both. Using this product, they gave a fully explicit combinatorial construction of D-regular graphs having spectral gap 1 − O(D − 1 3). In the same paper, they posed the open problem of whether a similar graph product could be used to achieve the almost optimal spectral gap 1 − O(D − 1 2). In this paper we propose a generalization of the zig-zag product that combines a large graph and several small graphs. The new product gives a better relation between the degree and the spectral gap of the resulting graph. We use the new product to give a fully explicit combinatorial construction of D-regular graphs having spectral gap 1 − D − 1 2 +o(1)