Cayley Graphs Without a Bounded Eigenbasis
Name
2005.04502.pdf
Description
Accepted version
Size
1.25 MB
Format
Adobe PDF
Checksum (MD5)
cbae6630f98f79aedf678ebb3ae6b637
Author(s)
Sah, Ashwin
Sawhney, Mehtaab
Zhao, Yufei
Date Issued
2020
Journal
International Mathematics Research Notices
Publisher
Oxford University Press (OUP)
Citation
Sah, Ashwin, Sawhney, Mehtaab and Zhao, Yufei. 2020. "Cayley Graphs Without a Bounded Eigenbasis." International Mathematics Research Notices, 2022 (8).
Version
Author's final manuscript
Abstract
Abstract
Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.
Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1093/IMRN/RNAA298
