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Abstract: A random $n$-lift of a base graph $G$ is its cover graph $H$ on the vertices$n\times VG$, where for each edge $u v$ in $G$ there is an independentuniform bijection $\pi$, and $H$ has all edges of the form $i,u,\pii,v$.A main motivation for studying lifts is understanding Ramanujan graphs, andnamely whether typical covers of such a graph are also Ramanujan.Let $G$ be a graph with largest eigenvalue $\lambda 1$ and let $ ho$ be thespectral radius of its universal cover. Friedman 2003 proved that every -new-eigenvalue of a random lift of $G$ is $O ho^{1-2}\lambda 1^{1-2}$ with highprobability, and conjectured a bound of $ ho+o1$, which would be tight byresults of Lubotzky and Greenberg 1995. Linial and Puder 2008 improvedFriedman-s bound to $O ho^{2-3}\lambda 1^{1-3}$. For $d$-regular graphs,where $\lambda 1=d$ and $ ho=2\sqrt{d-1}$, this translates to a bound of$Od^{2-3}$, compared to the conjectured $2\sqrt{d-1}$.Here we analyze the spectrum of a random $n$-lift of a $d$-regular graphwhose nontrivial eigenvalues are all at most $\lambda$ in absolute value. Weshow that with high probability the absolute value of every nontrivialeigenvalue of the lift is $O\lambda \vee ho \log ho$. This result istight up to a logarithmic factor, and for $\lambda \leq d^{2-3-\epsilon}$ itsubstantially improves the above upper bounds of Friedman and of Linial andPuder. In particular, it implies that a typical $n$-lift of a Ramanujan graphis nearly Ramanujan.

Autor: Eyal Lubetzky, Benny Sudakov, Van Vu

Fuente: https://arxiv.org/

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