Explicit two-sided unique-neighbor expanders
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We study the problem of constructing explicit sparse imbalanced bipartite unique-neighbor expanders. For large enough d_1 and d_2, we give a strongly explicit construction of an infinite family of (d_1,d_2)-biregular graph (assuming d_1 ≤ d_2) where all sets S with fewer than 1/d_1^3 fraction of vertices have Ω(d_1· |S|) unique-neighbors. Further, for each β∈(0,1), we give a construction with the additional property that the left side of each graph has roughly β fraction of the total number of vertices. Our work provides the first two-sided construction of imbalanced unique-neighbor expanders, meaning small sets contained in both the left and right side of the bipartite graph exhibit unique-neighbor expansion. Our construction is obtained from the “line product” of a large small-set edge expander and a sufficiently good constant-sized unique-neighbor expander, a product defined in the work of Alon and Capalbo.
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