On the Expansion of Group-Based Lifts

Abstract

A $k$-lift of an $n$-vertex base graph $G$ is a graph $H$ on $n\times k$ vertices, where each vertex $v$ of $G$ is replaced by $k$ vertices $v_1,\ldots,v_k$ and each edge $uv$ in $G$ is replaced by a matching representing a bijection $\pi_{uv}$ so that the edges of $H$ are of the form $(u_i,v_{\pi_{uv}(i)})$. Lifts have been investigated as a means to efficiently construct expanders. In this work, we study lifts obtained from groups and group actions. We derive the spectrum of such lifts via the representation theory principles of the underlying group. Our main results are 1. a uniform random lift by a cyclic group of order $k$ of any $n$-vertex $d$-regular base graph $G$, with the nontrivial eigenvalues of the adjacency matrix of $G$ bounded by $\lambda$ in magnitude, has the new nontrivial eigenvalues bounded by $\lambda+\mathcal{O}(\sqrt{d})$ in magnitude with probability $1-ke^{-\Omega(n/d^2)}$. The probability bounds as well as the dependency on $\lambda$ are almost optimal. As a special case, we obtain that there is a constant $c_1$ such that for every $k\leq 2^{c_1n/d^2}$, there exists a lift $H$ of every Ramanujan graph by a cyclic group of order $k$ such that $H$ is almost Ramanujan (nontrivial eigenvalues of the adjacency matrix at most $O(\sqrt{d})$ in magnitude). This result leads to a quasi-polynomial time deterministic algorithm to construct almost Ramanujan expanders; 2. there is a constant $c_2$ such that for every $k\geq 2^{c_2nd}$, there does not exist an abelian $k$-lift $H$ of any $n$-vertex $d$-regular base graph such that $H$ is almost Ramanujan. This can be viewed as an analogue of the well-known nonexpansion result for constant degree abelian Cayley graphs. Suppose $k_0$ is the order of the largest abelian group that produces expanding lifts. Our two results highlight lower and upper bounds on $k_0$ that are tight up to a factor of $d^3$ in the exponent, thus suggesting a threshold phenomenon.