Expansion properties of Cayley graphs of the alternating groups

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Abstract

Let C be a conjugacy class in the alternating group An, and let supp(C) be the number of nonfixed digits under the action of a permutation in C. For every 1 > δ > 0 and n ≥5 there exists a constant c = c(δ) > 0 such that if supp(C) ≥ δn then the undirected Cayley graph X(An, C) is a c expander. A family of such Cayley graphs with supp(C) = o(√n) is not a family of c-expanders. For every δ > 0, if supp(C) ≥ √3n then sets of vertices of order at most (1/2 - δ)(n - (n/supp(C)))! in X(An, C) expand. The proof of the last result combines spectral and representation theory techniques with direct combinatorial arguments.

Original languageEnglish
Pages (from-to)281-297
Number of pages17
JournalJournal of Combinatorial Theory. Series A
Volume79
Issue number2
DOIs
StatePublished - Aug 1997
Externally publishedYes

Bibliographical note

Funding Information:
* Partially sponsored by the Wolfson fellowship and the Hebrew University.

Funding

* Partially sponsored by the Wolfson fellowship and the Hebrew University.

FundersFunder number
Hebrew University of Jerusalem

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