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On the Second Eigenvalue and Random Walks in Random d-Regular Graphs (1993)
| Citations: | 80 - 10 self |
BibTeX
@MISC{Friedman93onthe,
author = {Joel Friedman},
title = {On the Second Eigenvalue and Random Walks in Random d-Regular Graphs},
year = {1993}
}
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Abstract
The main goal of this paper is to estimate the magnitude of the second largest eigenvalue in absolute value, 2 , of (the adjacency matrix of) a random d-regular graph, G. In order to do so, we study the probability that a random walk on a random graph returns to its originating vertex at the k-th step, for various values of k. Our main theorem about eigenvalues is that E fj 2 (G)j m g / 2 p 2d \Gamma 1 / 1 log d p 2d O ` 1 p d ' ! O ` d 3=2 log log n log n ' !m for any m 2 \Xi log n b p 2d \Gamma 1=2c= log d \Pi , where E f g denotes the expected value over a certain probablity space of 2d-regular graphs. It follows, for example, that for fixed d the second eigenvalue's magnitude is no more than 2 p 2d \Gamma 1 2 log d C 0 with probability 1 \Gamma n \GammaC for constants C and C 0 for sufficiently large n.
