arXiv 0911.41482009-11-21math.COmath.PR

提升Ramanujan图的谱

Spectra of lifted Ramanujan graphs

Eyal Lubetzky, Benny Sudakov, Van Vu

备注：: 34 pages, 4 figures

摘要：

基图$G$的随机$n$ -提升是其顶点$[n]\times V(G)$上的覆盖图$H$，其中对于$G$中的每条边$u v$，存在独立的一致双射$π$，并且$H$具有$(i,u),(π(i),v)$形式的所有边。研究升降机的一个主要动机是理解拉马努金图，即这样的图的典型覆盖是否也是拉马努金。设$G$是具有最大特征值$λ_1$的图，$ρ$是它的泛覆盖的谱半径。Friedman（2003）证明了$G$的随机提升的每一个“新”特征值都以高概率为$O(ρ^{1/2}λ_1^{1/2})$，并给出了$ρ+o(1)$的一个界，该界比Lubotzky和Greenberg（1995）的结果更紧. Linial和Puder（2008）改进了Friedman与$O(ρ^{2/3}λ_1^{1/3})$的结合。对于$d$ -正则图，其中$λ_1=d$和$ρ=2\sqrt{d-1}$，这转化为$O(d^{2/3})$的一个界限，相比于固定的$2\sqrt{d-1}$。在这里，我们分析了一个随机$n$ -提升的$d$ -正则图，其非平凡特征值都是在最多$λ$的绝对值的频谱。我们表明，以很高的概率，每一个非平凡的电梯特征值的绝对值是$O((λ\vee ρ) \log ρ)$。这个结果是紧密的对数因子，并为$λ\leq d^{2/3-ε}$，它大大提高了弗里德曼和Linial和Puder的上述上限。特别地，它意味着一个典型的$n$ -提升的Ramanujan图几乎是Ramanujan。

英文摘要：

A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices $[n]\times V(G)$, where for each edge $u v$ in $G$ there is an independent uniform bijection $π$, and $H$ has all edges of the form $(i,u),(π(i),v)$. A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan. Let $G$ be a graph with largest eigenvalue $λ_1$ and let $ρ$ be the spectral radius of its universal cover. Friedman (2003) proved that every "new" eigenvalue of a random lift of $G$ is $O(ρ^{1/2}λ_1^{1/2})$ with high probability, and conjectured a bound of $ρ+o(1)$, which would be tight by results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved Friedman's bound to $O(ρ^{2/3}λ_1^{1/3})$. For $d$-regular graphs, where $λ_1=d$ and $ρ=2\sqrt{d-1}$, this translates to a bound of $O(d^{2/3})$, compared to the conjectured $2\sqrt{d-1}$. Here we analyze the spectrum of a random $n$-lift of a $d$-regular graph whose nontrivial eigenvalues are all at most $λ$ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is $O((λ\vee ρ) \log ρ)$. This result is tight up to a logarithmic factor, and for $λ\leq d^{2/3-ε}$ it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical $n$-lift of a Ramanujan graph is nearly Ramanujan.

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