Expansion of random graphs: new proofs, new results

Abstract

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let $\Gamma$ be a random $d$-regular graph on $n$ vertices, and let $\lambda$ be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon (Combinatorica 6(2), 83–96, 1986) that a random $d$-regular graph is “almost Ramanujan”, in the following sense: for every $\varepsilon >0$, $\lambda <2\sqrt{d-1}+\varepsilon$ asymptotically almost surely. Friedman famously presented a proof of this conjecture in Friedman (Memoirs of the AMS 910, 2008). Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random $d$-regular graph satisfies $\lambda <2\sqrt{d-1}+1$ a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For $d$ even, a $d$-regular graph on $n$ vertices is an $n$-covering space of a bouquet of $d/2$ loops. More generally, fixing an arbitrary base graph $\Omega$, we study the spectrum of $\Gamma$, a random $n$-covering of $\Omega$. Let $\lambda$ be the largest absolute value of a non-trivial eigenvalue of $\Gamma$. Extending Alon’s conjecture to this more general model, Friedman (Duke Math J 118(1),19–35, 2003) conjectured that for every $\varepsilon >0,$ a.a.s. $\lambda <\rho +\varepsilon$, where $\rho$ is the spectral radius of the universal cover of $\Omega$. When $\Omega$ is regular we get a bound of $\rho +0.84$, and for an arbitrary $\Omega$, we prove a nearly optimal upper bound of $\sqrt{3}\rho$. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

Appendices

Appendices

A Contiguity and related models of random graphs

1.1 A.1 Random $d$-regular graphs

In this paper, the statement of Theorem 1.1 is first proved for the permutation model of random $d$-regular graphs with $d$ even. We then derive Theorem 1.1, stated for the uniform distribution on all $d$-regular simple graphs on $n$ vertices with $d$ even or odd, using results of Wormald [61] and Greenhill et al. [25]. These works show the contiguity (see footnote 3) of different models of random regular graphs.

In particular, they describe the following model: consider $dn$ labeled points, with $d$ points in each of $n$ buckets, and take a random perfect matching of the points. Letting the buckets be vertices and each pair represent an edge, one obtains a random $d$-regular graph. This model is denoted $\mathcal{G}_{n,d}^{*}$. It is shown [25, Theorem 1.3] that $\mathcal{G}_{n,d}^{*}$ is contiguous to the permutation model $\mathcal{P}_{n,d}$ (for $d$ even). If $\Gamma$ is a random $d$-regular graph in $\mathcal{G}_{n,d}^{*}$, the event that $\Gamma$ is a simple graph (with no loops nor multiple edges) has positive probability, bounded away from 0. Moreover, within this event, simple graphs are distributed uniformly.²⁵ Thus, for even values of $d$, Theorem 1.1 follows from the corresponding result for the permutation model. The derivation of the odd case also uses contiguity results, as explained in Sect. 6.2.

1.2 A.2 Random $d$-regular bipartite graphs

As an immediate corollary from Theorem 1.5 we deduced that a random $d$-regular bipartite graph is “nearly Ramanujan” in the sense that besides its two trivial eigenvalues $\pm d$, all other eigenvalues are at most $2\sqrt{d-1}+0.84$ in absolute value a.a.s. (Corollary 1.6). Our proof works in the model $C_{n,\Omega }$ (here $\Omega$ is the graph with $2$ vertices and $d$ parallel edges connecting them). However, by the results of [9], the probability that our graph has no multiple edges is bounded away from zero (asymptotically it is $e^{-\left( {\begin{array}{c}d\\ 2\end{array}}\right) }$). Thus, our result applies also to the model of $d$ random disjoint perfect matchings between two sets of $n$ vertices. This model, in turn, is contiguous to the uniform model of bipartite (vertex-labeled) $d$-regular simple graphs (for $d\ge 3$: see [44, Section 4]²⁶), so our result applies in the latter model as well.

1.3 A.3 Random coverings of a fixed graph

In Theorem 1.4 we consider random $n$-coverings of a fixed graph $\Omega$ in the model $\mathcal{C}_{n,\Omega }$, where a uniform random permutation is generated for every edge of $\Omega$. An equivalent model is attained if we cover some spanning tree of $\Omega$ by $n$ disjoint copies and then choose a random permutation for every edge outside the tree (that is, the same automorphism-types of non-labeled graphs are obtained with the same distribution). In fact, picking a basepoint $\otimes \in V\left( \Omega \right)$, there is yet another description for this model: The classification of $n$-sheeted coverings of $\Omega$ by the action of $\pi _{1}\left( \Omega ,\otimes \right)$ on the fiber $\left\{ \otimes \right\} \times \left[ n\right]$ above $\otimes$ shows that $\mathcal{C}_{n,\Omega }$ is equivalent to choosing uniformly at random an action of the free group $\pi _{1}\left( \Omega ,\otimes \right)$ on $\left\{ \otimes \right\} \times \left[ n\right]$.

A different but related model uses the classification of connected, pointed coverings of $\left( \Omega ,\otimes \right)$ by the corresponding subgroups of $\pi _{1}\left( \Omega ,\otimes \right)$. A random $n$-covering is thus generated by choosing a random subgroup of index $n$. However, it seems that this model is contiguous to $\mathcal{C}_{n,\Omega }$ if $\mathrm {rk}\left( \Omega \right) \ge 2$ (note that the random covering $\Gamma$ in $\mathcal{C}_{n,\Omega }$ is a.a.s. connected provided that $\mathrm {rk\left( \Omega \right) \ge 2}$). Indeed, the only difference is that in the new model, the probability of every connected graph $\Gamma$ from $\mathcal{C}_{n,\Omega }$ is proportional to $\frac{1}{\left| \mathrm {Aut}\left( \Gamma \right) \right| }$. When $\mathrm {rk\left( \Omega \right) \ge 2}$, it seems that a.a.s. $\left| \mathrm {Aut}\left( \Gamma \right) \right| =1$, which would show that our result applies to this model as well.

Finally, there is another natural model that comes to mind: given a periodic infinite tree, namely a tree that covers some finite graph, one can consider a random (simple) graph $\Gamma$ with $n$ vertices covered by this tree (with uniform distribution among all such graphs with $n$ vertices, for suitable $n$’s only). One can then analyze $\lambda \left( \Gamma \right)$, the largest absolute value of an eigenvalue besides²⁷ ${\mathfrak { pf}}\left( \Gamma \right)$. (This generalizes the uniform model on $d$-regular graphs.) Occasionally, all the quotients of some given periodic tree $T$ cover the same finite “minimal” graph $\Omega$. Interestingly, Lubotzky and Nagnibeda [36] showed that there exist such $T$’s with a minimal quotient $\Omega$ which is not Ramanujan (in the sense that $\lambda \left( \Omega \right)$ is strictly larger than $\rho \left( T\right)$, the spectral radius of $T$). Since all the quotients of $T$ inherit the eigenvalues of $\Omega$, their $\lambda \left( \cdot \right)$ is also bounded away from $\rho \left( T\right)$ (from above). Hence, the corresponding version of Conjecture 1.3 is false in this general setting.

B Spectral expansion of non-regular graphs

In this section we provide some background on the theory of expansion of irregular graphs, describing how spectral expansion is related to other measurements of expansion (combinatorial expansion, random walks and mixing). This further motivates the claim that Theorem 1.4 shows that if the base graph $\Omega$ is a good (nearly optimal) expander, then a.a.s. so are its random coverings. We would like to thank Ori Parzanchevski for his valuable assistance in writing this appendix.

The spectral expansion of a (non-regular) graph $\Gamma$ on $m$ vertices is measured by some function on its spectrum, and most commonly by the spectral gap: the difference between the largest eigenvalue and the second largest. As mentioned above, it is not apriori clear which operator best describes in spectral terms the properties of the graph. There are three main candidates (see, e.g. [28]), all of which are bounded²⁸, self-adjoint operators and so have real spectrum:

• (1) The adjacency operator $A_{\Gamma }$ on $\left( \ell ^{2}\left( V\left( \Gamma \right) \right) ,1\right)$ ²⁹:

  $$\begin{aligned} (A_{\Gamma }f)(v)=\sum _{w\sim v}f\left( w\right) \end{aligned}$$

  If $\Gamma$ is finite this operator is represented in the standard basis by the adjacency matrix, and its spectral radius is the Perron-Frobenius eigenvalue ${\mathfrak { pf}}\left( \Gamma \right)$. The spectrum in this case is

  $$\begin{aligned} {\mathfrak { pf}}\left( \Gamma \right) =\lambda _{1}\ge \lambda _{2}\ge \cdots \ge \lambda _{m}\ge -{\mathfrak { pf}}\left( \Gamma \right) , \end{aligned}$$

  and the spectral gap is ${\mathfrak { pf}}\left( \Gamma \right) -\lambda \left( \Gamma \right)$, where $\lambda \left( \Gamma \right) =\max \left\{ \lambda _{2},-\lambda _{n}\right\}$ ³⁰. The spectrum of $A_{\Gamma }$ was studied in various works, for instance [20, 24, 33, 36].

• (2) The averaging Markov operator $M_{\Gamma }$ on $\left( \ell ^{2}\left( V\left( \Gamma \right) \right) ,\deg \left( \cdot \right) \right)$ ³¹:

  $$\begin{aligned} (M_{\Gamma }f)(v)=\frac{1}{\deg \left( v\right) }\sum _{w\sim v}f\left( w\right) \end{aligned}$$

  This operator is given by $D_{\Gamma }^{-1}A_{\Gamma }$, and its spectrum is contained in $\left[ -1,1\right]$. The eigenvalue $1$ corresponds to locally-constant functions when $\Gamma$ is finite, and in this case the spectrum is

  $$\begin{aligned} 1=\mu _{1}\ge \mu _{2}\ge \cdots \ge \mu _{m}\ge -1. \end{aligned}$$

  The spectral gap is then $1-\mu \left( \Gamma \right)$ here $\mu \left( \Gamma \right) =\max \left\{ \mu _{2},-\mu _{m}\right\}$. Up to a possible affine transformation, the spectrum of $M_{\Gamma }$ is the same as the spectrum of the simple random walk operator ($A_{\Gamma }D_{\Gamma }^{-1}$) or of one of the normalized Laplacian operators ($I-A_{\Gamma }D_{\Gamma }^{-1}$ or $I-D_{\Gamma }^{-1/2}A_{\Gamma }D_{\Gamma }^{-1/2}$). This spectrum is considered for example in [17, 27, 58].

• (3) The Laplacian operator $\Delta _{\Gamma }^{+}$ on $\left( \ell ^{2}\left( V\left( \Gamma \right) \right) ,1\right)$:

  $$\begin{aligned} \left( \Delta _{\Gamma }^{+}f\right) \left( v\right) =\deg \left( v\right) f\left( v\right) -\sum _{w\sim v}f\left( w\right) \end{aligned}$$

  The Laplacian equals $D_{\Gamma }-A_{\Gamma }$, where $D_{\Gamma }$ is the diagonal operator $\left( D_{\Gamma }f\right) \left( v\right) =\deg \left( v\right) \cdot f\left( v\right)$. The entire spectrum is non-negative, with $0$ corresponding to locally-constant functions when $\Gamma$ is finite. In the finite case, the spectrum is

  $$\begin{aligned} 0=\nu _{1}\le \nu _{2}\le \cdots \le \nu _{m}, \end{aligned}$$

  the spectral gap being $\nu _{2}-\nu _{1}=\nu _{2}$. The Laplacian operator is studied e.g. in [3].

For a regular graph $\Gamma$, all different operators are identical up to an affine shift. However, in the general case there is no direct connection between the three different spectra. In this paper we consider the spectra of $A_{\Gamma }$ and of $M_{\Gamma }$. At this point we do not know how to extend our results to the Laplacian operator $\Delta _{\Gamma }^{+}$.

The spectrum of all three operators is closely related to different notions of expansion in graphs. The adjacency operator, for example, has the following version of the expander mixing lemma: for every two subsets $S,T\subseteq V\left( \Gamma \right)$ (not necessarily disjoint), one has

$$\begin{aligned} \left| E\left( S,T\right) -{\mathfrak { pf}}\left( \Gamma \right) \mathrm {vol}_{{\mathfrak { pf}}}\left( S\right) \mathrm {vol}_{{\mathfrak { pf}}}\left( T\right) \right| \le \lambda \left( \Gamma \right) \frac{\sqrt{\left| S\right| \cdot \left| T\right| }}{m}, \end{aligned}$$

where $\mathrm {vol}_{{\mathfrak { pf}}}\left( S\right) =\left\langle \mathfrak {1}_{S},f_{{\mathfrak { pf}}}\left( \Gamma \right) \right\rangle$ and $f_{{\mathfrak { pf}}}\left( \Gamma \right)$ is the (normalized) Perron-Frobenius eigenfunction. This is particularly useful in the $\mathcal{C}_{n,\Omega }$ model since the $f_{{\mathfrak { pf}}}\left( \Gamma \right)$ is easily obtained from the Perron-Frobenius eigenfunction of $\Omega$ by

$$\begin{aligned} f_{{\mathfrak { pf}}}\left( \Gamma \right) =\frac{1}{\sqrt{n}}f_{{\mathfrak { pf}}}\left( \Omega \right) \circ \pi . \end{aligned}$$

In the $d$-regular case, this amounts to the usual mixing lemma: $\Big |E(S,T)-d\frac{\left| S\right| \cdot \left| T\right| }{m}\Big |\le \lambda \left( \Gamma \right) \sqrt{\left| S\right| \cdot \left| T\right| }$. If one takes $T=V{\setminus } S$, one can attain a bound on the Cheeger constant of $\Gamma$ [see (9.1)].

As for the averaging Markov operator, it is standard that $\mu \left( \Gamma \right)$ controls the speed in which a random walk converges to the stationary distribution. In addition, if one defines $\mathrm {deg}\left( S\right)$ to denote the sum of degrees of the vertices in $S$, then

$$\begin{aligned} \left| E\left( S,T\right) -\frac{\mathrm {deg}\left( S\right) \mathrm {deg}\left( T\right) }{2\left| E\left( \Gamma \right) \right| }\right| \le \mu \left( \Gamma \right) \sqrt{\mathrm {deg}\left( S\right) \mathrm {deg}\left( T\right) }. \end{aligned}$$

Moreover, consider the conductance of $\Gamma$

$$\begin{aligned} \phi \left( \Gamma \right) =\min _{\begin{array}{c} \emptyset \ne S\subseteq V \\ \mathrm {deg}\left( S\right) \le \frac{\mathrm {deg}\left( V\right) }{2} \end{array}} \frac{\left| E\left( S,V{\setminus } S\right) \right| }{\mathrm {deg}\left( S\right) }. \end{aligned}$$

Then the following version of the Cheeger inequality holds [58, Lemmas 2.4, 2.6]:

$$\begin{aligned} \frac{\phi ^{2}\left( \Gamma \right) }{2}\le 1-\mu _{2}\le 2\phi \left( \Gamma \right) . \end{aligned}$$

Finally, the spectrum of the Laplacian operator is related to the standard Cheeger Constant of $\Gamma$, defined as

$$\begin{aligned} h\left( \Gamma \right) =\min _{\begin{array}{c} \emptyset \ne S\subseteq V \\ \left| S\right| \le \frac{\left| V\right| }{2} \end{array}} \frac{\left| E\left( S,V{\setminus } S\right) \right| }{\left| S\right| }. \end{aligned}$$

(9.1)

By the so-called “discrete Cheeger inequality” [3]:

$$\begin{aligned} \frac{h^{2}\left( \Gamma \right) }{2k}\le \nu _{2}\le 2h\left( \Gamma \right) \end{aligned}$$

with $k$ being the largest degree of a vertex. In addition, one has a variation on the mixing lemma for $\Delta _{\Gamma }^{+}$ as well [50, Thm 1.4].