Discrete Mathematics & Theoretical Computer Science
| author: | Loh, Po-Shen and Schulman, Leonard J. |
|---|---|
| title: | Improved Expansion of Random Cayley Graphs |
| keywords: | expander graphs, Cayley graphs, second eigenvalue, logarithmic generators |
| abstract: | In Random Cayley Graphs and Expanders, N. Alon and Y. Roichman proved that for every ε > 0
there is a finite c(ε) such that for any
sufficiently large group G, the expected value of the
second largest (in absolute value) eigenvalue of the normalized
adjacency matrix of the Cayley graph with respect to
c(ε) log |G| random elements is less than
ε. We reduce the number of elements to
c(ε)log D(G) (for the same c),
where D(G) is the sum of the dimensions of the
irreducible representations of G. In sufficiently
non-abelian families of groups (as measured by these dimensions), log D(G) is asymptotically (1/2)log|G|. As is well known, a small eigenvalue implies large graph expansion (and conversely); see Tanner84 and AlonMilman84-2,AlonMilman84-1. For any specified eigenvalue or expansion, therefore, random Cayley graphs (of sufficiently non-abelian groups) require only half as many edges as was previously known.
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| reference: | Loh, Po-Shen and Schulman, Leonard J. (2004), Improved Expansion of Random Cayley Graphs, Discrete Mathematics and Theoretical Computer Science 6, pp. 523-528 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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