Random Graph Lifts

Let G be any connected graph, and let λ1 and ρ denote the spectral radius of G and the universal cover of G, respectively. In [LP09], Linial and Puder have shown that almost every n-lift of G has all of its new eigenvalues bounded by O(λ 1/3 1 ρ 2/3). Friedman had conjectured that this bound can be improved to ρ+ on(1) (e.g., see [Fri03, HLW06]). In [LP09], Linial and Puder have formulated two new categorizations of formal words, namely φ and β, which assigns a non-negative integer or infinity to each word. They have shown that for every word w, φ(w) = 0 iff β(w) = 0, and φ(w) = 1 iff β(w) = 1. They have conjectured that φ(w) = β(w) for every word w, and they have run extensive numerical simulations that suggest that this conjecture is true. This conjecture, if proven true, gives us a promising approach to proving a slightly weaker version of Friedman’s conjecture, namely the bound O(ρ) (see [LP09]). In this paper, we show that φ(w) = 2 iff β(w) = 2 for every word w. We also discuss possible strategies for proving φ(w) = 3 iff β(w) = 3.