再论phi = beta猜想与随机图提升的特征值
More on the phi = beta Conjecture and Eigenvalues of Random Graph Lifts
摘要：
设$G$是一个连通图，$λ_1$和$ρ$分别表示$G$的谱半径和$G$的泛覆盖.在\cite{Fri03}中，Friedman证明了几乎$G$的每个$n$ -提升都有其所有新的特征值以$O(λ_1^{1/2}ρ^{1/2})$为界。在\cite{LP10}中，Linial和Puder将该结合改进为$O(λ_1^{1/3}ρ^{2/3})$。弗里德曼已经证实，这个界限实际上可以改进为$ρ+ o_n(1)$（例如，参见\cite{Fri03,HLW06}）。在\cite{LP10}中，Linial和Puder制定了两种新的形式词分类，即$ϕ$和$β$，它们为每个词分配非负整数或无穷大。他们已经证明，对于每个单词$w$，$ϕ(w) = 0$当且仅当$β(w) = 0$，以及$ϕ(w) = 1$当且仅当$β(w) = 1$。他们证明了每个单词都有$ϕ(w) = β(w)$ $w$，并进行了大量的数值模拟，强烈表明这一猜想是正确的。这个猜想，如果被证明是真的，给我们一个非常有前途的方法来证明一个稍微弱版本的弗里德曼猜想，即绑定$O(ρ)$的新的特征值（见\cite{LP10}）。在本文中，我们取得了进一步的进展，证明这一重要的猜想，表明$ϕ(w) = 2$当且仅当$β(w) = 2$为每个字$w$。
英文摘要：
Let $G$ be a connected graph, and let $λ_1$ and $ρ$ 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 $O(λ_1^{1/2}ρ^{1/2})$. In \cite{LP10}, Linial and Puder have improved this bound to $O(λ_1^{1/3}ρ^{2/3})$. Friedman had conjectured that this bound can actually be improved to $ρ+ o_n(1)$ (e.g., see \cite{Fri03,HLW06}). In \cite{LP10}, Linial and Puder have formulated two new categorizations of formal words, namely $ϕ$ and $β$, which assign 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 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 $O(ρ)$ on the new eigenvalues (see \cite{LP10}). In this paper, we make further progress towards proving this important conjecture by showing that $ϕ(w) = 2$ iff $β(w) = 2$ for every word $w$.
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