Discrete Mathematics & Theoretical Computer Science

Volume 6 n° 2 (2004), pp. 523-528

author: Loh, Po-Shen and Schulman, Leonard J. Improved Expansion of Random Cayley Graphs expander graphs, Cayley graphs, second eigenvalue, logarithmic generators 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. If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files. Loh, Po-Shen and Schulman, Leonard J. (2004), Improved Expansion of Random Cayley Graphs, Discrete Mathematics and Theoretical Computer Science 6, pp. 523-528 For a corresponding BibTeX entry, please consider our BibTeX-file. dm060222.ps.gz (35 K) dm060222.ps (87 K) dm060222.pdf (67 K)

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