Let $G$ be a connected graph, and let $��_1$ and $��$ denote the spectral radius of $G$ and the universal cover of $G$, respectively. In \cite{Fri03}, Friedman has shown that almost every $n$-lift of $G$ has all of its new eigenvalues bounded by $O(��_1^{1/2}��^{1/2})$. In \cite{LP10}, Linial and Puder have improved this bound to $O(��_1^{1/3}��^{2/3})$. Friedman had conjectured that this bound can actually be improved to $��+ o_n(1)$ (e.g., see \cite{Fri03,HLW06}). In \cite{LP10}, Linial and Puder have formulated two new categorizations of formal words, namely $��$ and $��$, which assign a non-negative integer or infinity to each word. They have shown that for every word $w$, $��(w) = 0$ iff $��(w) = 0$, and $��(w) = 1$ iff $��(w) = 1$. They have conjectured that $��(w) = ��(w)$ for every word $w$, and have run extensive numerical simulations that strongly suggest that this conjecture is true. This conjecture, if proven true, gives us a very promising approach to proving a slightly weaker version of Friedman's conjecture, namely the bound $O(��)$ on the new eigenvalues (see \cite{LP10}). In this paper, we make further progress towards proving this important conjecture by showing that $��(w) = 2$ iff $��(w) = 2$ for every word $w$.

24 pages