We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion. Specifically, we show that the one-sided lossless expanders constructed by Kalev and Ta-Shma (RANDOM'22)--that are based on multiplicity codes introduced by Kopparty, Saraf, and Yekhanin (STOC'11)--are, in fact, two-sided lossless expanders. Using our unbalanced bipartite expander, we easily obtain lossless (non-bipartite) expander graphs on $N$ vertices with polynomial degree and a free group action. As far as we know, this is the first explicit construction of lossless (non-bipartite) expanders with $N$ vertices and degree $\ll N$.

翻译：我们首次在非平衡设置（左侧节点远多于右侧节点的二分图）中给出了双向无损扩展图的显式构造。在本工作之前，所有已知的非平衡设置显式构造仅能实现单向无损扩展。具体而言，我们证明了 Kalev 与 Ta-Shma (RANDOM'22) 基于 Kopparty、Saraf 和 Yekhanin (STOC'11) 所引入的多重码构造的单向无损扩展图实际上也是双向无损扩展图。利用我们的非平衡二分扩展图，我们可以轻松获得具有多项式度数和自由群作用的 $N$ 个顶点上的无损（非二分）扩展图。据我们所知，这是首个具有 $N$ 个顶点且度 $\ll N$ 的无损（非二分）扩展图的显式构造。