On the girth of random Cayley graphs
Probability
2011-11-10 v1 Group Theory
Abstract
We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/2 and that random d-regular Cayley graphs of simple algebraic groups over F_q asymptotically almost surely have girth at least log_{d-1}|G|/dim(G). For the symmetric p-groups the girth is between log log |G| and (log|G|)^alpha with alpha<1. Several conjectures and open questions are presented.
Cite
@article{arxiv.0707.1833,
title = {On the girth of random Cayley graphs},
author = {Alex Gamburd and Shlomo Hoory and Mehrdad Shahshahani and Aner Shalev and Balint Virag},
journal= {arXiv preprint arXiv:0707.1833},
year = {2011}
}
Comments
20 pages