TY - GEN
T1 - Tight Bounds for the Zig-Zag Product
AU - Cohen, Gil
AU - Cohen, Itay
AU - Maor, Gal
N1 - Publisher Copyright:
© 2024 IEEE.
PY - 2024
Y1 - 2024
N2 - The Zig-Zag product of two graphs, Z= GzH, was introduced in the seminal work of Reingold, Vadhan, and Wigderson (Ann. of Math. 2002) and has since become a pivotal tool in theoretical computer science. The classical bound, which is used throughout, states that the spectral expansion of the Zig-Zag product can be bounded roughly by the sum of the spectral expansions of the individual graphs, Ω z≤ΩH+ΩG. In this work we derive, for every (vertex-transitive) c-regular graph H on d vertices, a tight bound for Ω z by taking into account the entire spectrum of H. Our work reveals that the bound, which holds for every graph G, is precisely the minimum value of the function (Equation Presented) in the domain (c2, ∞), where H(x) is the characteristic polynomial of H2. As a consequence, we establish that Zig-Zag products are indeed intrinsically quadratic away from being Ramanujan. We further prove tight bounds for the spectral ex-pansion of the more fundamental replacement product. Our lower bounds are based on results from analytic combinatorics, and we make use of finite free probability to prove their tightness. In a broader context, our work uncovers intriguing links between the two fields and these well-studied graph operators.
AB - The Zig-Zag product of two graphs, Z= GzH, was introduced in the seminal work of Reingold, Vadhan, and Wigderson (Ann. of Math. 2002) and has since become a pivotal tool in theoretical computer science. The classical bound, which is used throughout, states that the spectral expansion of the Zig-Zag product can be bounded roughly by the sum of the spectral expansions of the individual graphs, Ω z≤ΩH+ΩG. In this work we derive, for every (vertex-transitive) c-regular graph H on d vertices, a tight bound for Ω z by taking into account the entire spectrum of H. Our work reveals that the bound, which holds for every graph G, is precisely the minimum value of the function (Equation Presented) in the domain (c2, ∞), where H(x) is the characteristic polynomial of H2. As a consequence, we establish that Zig-Zag products are indeed intrinsically quadratic away from being Ramanujan. We further prove tight bounds for the spectral ex-pansion of the more fundamental replacement product. Our lower bounds are based on results from analytic combinatorics, and we make use of finite free probability to prove their tightness. In a broader context, our work uncovers intriguing links between the two fields and these well-studied graph operators.
KW - combinatorics
KW - pseudorandomness and derandomization
KW - spectral methods
UR - http://www.scopus.com/inward/record.url?scp=85213013573&partnerID=8YFLogxK
U2 - 10.1109/FOCS61266.2024.00094
DO - 10.1109/FOCS61266.2024.00094
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AN - SCOPUS:85213013573
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 1470
EP - 1499
BT - Proceedings - 2024 IEEE 65th Annual Symposium on Foundations of Computer Science, FOCS 2024
PB - IEEE Computer Society
T2 - 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2024
Y2 - 27 October 2024 through 30 October 2024
ER -