TY - GEN
T1 - Explicit Two-Sided Unique-Neighbor Expanders
AU - Hsieh, Jun Ting
AU - McKenzie, Theo
AU - Mohanty, Sidhanth
AU - Paredes, Pedro
N1 - Publisher Copyright:
© 2024 Owner/Author.
PY - 2024/6/10
Y1 - 2024/6/10
N2 - 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 and the routed product of previous well-known works. 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 previously known results, 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 (d1,d2)-biregular graphs (Gn)n≥ 1 (for large enough d1 and d2) where all sets S with fewer than a small constant fraction of vertices have ω(d1· |S|) unique-neighbors (assuming d1 ≤ d2). Additionally, we can also guarantee that subsets of vertices of size up to exp(ω(√log|V(Gn)|)) expand losslessly.
AB - 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 and the routed product of previous well-known works. 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 previously known results, 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 (d1,d2)-biregular graphs (Gn)n≥ 1 (for large enough d1 and d2) where all sets S with fewer than a small constant fraction of vertices have ω(d1· |S|) unique-neighbors (assuming d1 ≤ d2). Additionally, we can also guarantee that subsets of vertices of size up to exp(ω(√log|V(Gn)|)) expand losslessly.
KW - Algebraic expanders
KW - Lossless expanders
KW - Unique-neighbor expanders
UR - http://www.scopus.com/inward/record.url?scp=85196658653&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85196658653&partnerID=8YFLogxK
U2 - 10.1145/3618260.3649705
DO - 10.1145/3618260.3649705
M3 - Conference contribution
AN - SCOPUS:85196658653
T3 - Proceedings of the Annual ACM Symposium on Theory of Computing
SP - 788
EP - 799
BT - STOC 2024 - Proceedings of the 56th Annual ACM Symposium on Theory of Computing
A2 - Mohar, Bojan
A2 - Shinkar, Igor
A2 - O�Donnell, Ryan
PB - Association for Computing Machinery
T2 - 56th Annual ACM Symposium on Theory of Computing, STOC 2024
Y2 - 24 June 2024 through 28 June 2024
ER -