## A proof of Alon’s second eigenvalue conjecture (2003)

Citations: | 91 - 1 self |

### BibTeX

@INPROCEEDINGS{Friedman03aproof,

author = {Joel Friedman},

title = {A proof of Alon’s second eigenvalue conjecture},

booktitle = {},

year = {2003},

pages = {720--724}

}

### Years of Citing Articles

### OpenURL

### Abstract

A d-regular graph has largest or first (adjacency matrix) eigenvalue λ1 = d. Consider for an even d ≥ 4, a random d-regular graph model formed from d/2 uniform, independent permutations on {1,...,n}. We shall show that for any ɛ>0 we have all eigenvalues aside from λ1 = d are bounded by 2 √ d − 1 +ɛwith probability 1 − O(n−τ), where τ = ⌈ � √ d − 1+1 � /2⌉−1. We also show that this probability is at most 1 − c/nτ ′, for a constant c and a τ ′ that is either τ or τ +1 (“more often ” τ than τ + 1). We prove related theorems for other models of random graphs, including models with d odd. These theorems resolve the conjecture of Alon, that says that for any ɛ>0andd, the second largest eigenvalue of “most ” random dregular graphs are at most 2 √ d − 1+ɛ (Alon did not specify precisely what “most ” should mean or what model of random graph one should take). 1