An expander is a graph which is sparse but at the same time highly connected.
Expanders are remarkable graphs which have many applications in mathematics and computer science. In the last forty years they were subject of a huge amount of research.
Here we will use them to show that Bourgain’s theorem is tight in the sense that the shortest path metric on expander graphs can only be embedded into Euclidean space with distortionΩplognq.
For this we start by defining the edge expansion ratio. Although this defini-tion gives some intuidefini-tion how expander graphs look like it is frequently much easier to work with expanders algebraically using spectral properties of their adjacency matrix. These spectral properties will then be useful for proving that expanders embed rather badly into Euclidean space.
11.4.1 Edge expansion
LetG “ pV, Eq be a graph. We assume that in Gevery vertex has exactly d neighbors, i.e. thatGisd-regular. LetSĎV be a subset of the vertices and let S“VzSbe its complement. Theedge boundaryofSis
BS“ ttu, vu PE:uPS, vPSu.
For the edges which stay inSorSdefine
EpSq “ ttu, vu PE:u, vPSu, EpSq “ ttu, vu PE:u, vPSu.
Definition 11.4.1. Theedge expansion ratioof a graphGis hpGq “ min
SĎV:|S|ď|V|{2
|BS|
|S|
Definition 11.4.2. Letdě3 be an integer. A family ofd-regular graphsGn “ pVn, Enqwith|Vn| Ñ 8 whenn tends to infinity is called afamily ofd-regular expander graphsif there existsą0withhpGnq ą.
In the following two sections we will prove a fundamental inequality due to Dodziuk (1984) and independently Alon and Milman (1985) and Alon (1986).
It relates the edge expansion ratiohpGqof ad-regular graph with thespectral gapof a graph, the differenced´λ2between the largest and the second largest eigenvalue of its adjacency matrix
d´λ2
2 ďhpGq ďa
2dpd´λ2q
This shows thatGn is a family ofd-regular expander graphs if and only if there exists aną0so thatd´λ2pGnq ąfor alln.
11.4.2 Large spectral gap implies high expansion
Theorem 11.4.3. LetG “ pV, Eqbe a connected, d-regular graph. Letλ1 “d andλ2be the largest and the second largest eigenvalue of the adjacency matrix of G. Then,
d´λ2
2 ďhpGq.
Proof. The largest eigenvalue of the adjacency matrixAof thed-regular graphG equalsdand the corresponding eigenvector is the all-ones vectore(see Exercise 11.2). So the second largest eigenvalueλ2ofAis given by
λ2“ max
fPRVzt0u,fKe
fTAf fTf
because of the Rayleigh principle. If we would find a vectorf which is perpen-dicular toeso that
fTAf
fTf ěd´2hpGq
holds, then we would prove the desired inequality. LetS ĎV be a set attaining the edge expansion ratio
hpGq “ |BS|
|S| , with|S| ď |V|{2.
Define the vector
f “ |S|χS´ |S|χS PRV
where χS P RV denotes the characteristic vector of the set S. This vector is perpendicular toe. The denominator of the Rayleigh quotient equals
fTAf “ 2 ÿ
tu,vuPE
fpuqfpvq
“ 2`
|EpSq||S|2` |EpSq||S|2´ |S||S||BS|˘
“ pd|S| ´ |BS|q|S|2` pd|S| ´ |BS|q|S|2´2|S||S||BS|
“ dp|S| ` |S|q|S||S| ´ p|S| ` |S|q2|BS|
“ d|V||S||S| ´ |V|2|BS|,
where we first split the sum intotu, vu PEpSq,tu, vu P EpSq, andtu, vu P BS, and then use the identities
d|S| “2|EpSq| ` |BS|, d|S| “2|EpSq| ` |BS|, |V| “ |S| ` |S|.
The numerator of the Rayleigh quotient equals
fTf “ |S|2|S| ` |S|2|S| “ |S||S|p|S| ` |S|q “ |V||S||S|.
Together, fTAf
fTf “ d|V||S||S| ´ |V|2|BS|
|V||S||S| “d´n|BS|
|S||S| ěd´2hpGq, where we use thathpGq “ |BS|{|S|and|S| ě |V|{2.
11.4.3 High expansion implies large spectral gap
Theorem 11.4.4. LetG “ pV, Eqbe a connected, d-regular graph. Letλ1 “d andλ2be the largest and the second largest eigenvalue of the adjacency matrix of G. Then,
hpGq ďa
2dpd´λ2q.
Proof. Letgbe an eigenvector of the adjacency matrixAofGcorresponding to λ2. Sinceg is perpendicular to the all-ones vector, the vectorg has positive as well as negative entries. Definef PRV by
fpuq “
"
gpuq ifgpuq ą0, 0 otherwise.
Let S “ tu P V : fpuq ‰ 0u be the support of f. We may assume that S has at most|V|{2vertices, otherwise we would replace the eigenvectorgby its negative´g.
The theorem will follow once we prove the inequalities hpGq2
2d ď fTLf
fTf ďd´λ2 (11.1)
for the Laplacian matrixL“dI´Aof thed-regular graphG.
The upper bound in(11.1) is (relatively) easy: ForuPSwe have pLfqpuq “ dfpuq ´ ÿ
vPV:tu,vuPE
fpvq
“ dgpuq ´ ÿ
vPS:tu,vuPE
gpvq ď dgpuq ´ ÿ
vPV:tu,vuPE
gpvq
“ pd´λ2qgpuq.
Becausefpuq “0wheneveruRSwe arrive at fTLf “ ÿ
uPV
fpuqpLfqpuq ď pd´λ2qÿ
uPS
gpuq2“ pd´λ2qfTf.
The lower bounds in(11.1) is harder and needs more work and ingenuity.
Some preparation: Let us label the vertices ofGby1, . . . ,|V|so that fp1q ěfp2q ě. . .ěfp|V|q.
Direct the edges of the graphG(arbitrarily) and defineKPRVˆEby
Kpu, eq “
$
&
%
`1 if edgeeenters vertexu,
´1 if edgeeexits vertexu, 0 otherwise.
Then one hasL“KKT. Define the quantity
B“ ÿ
tu,vuPE
|fpuq2´fpvq2|.
We shall prove
hpGqfTf ďBď
? 2d
b
pKfqTKfa
fTf , (11.2)
which implies the lower bound in (11.1) becausefTLf “ pKfqTpKfq.
The upper bound in (11.2) follows from Cauchy-Schwarz
B “ ÿ
tu,vuPE
|fpuq2´fpvq2|
“ ÿ
tu,vuPE
|fpuq `fpvq| ¨ |fpuq ´fpvq|
ď
d ÿ
pfpuq `fpvqq2¨ d ÿ
pfpuq ´fpvqq2
and by
d ÿ
tu,vuPE
pfpuq ´fpvqq2“ b
pKfqTKf as well by
d ÿ
tu,vuPE
pfpuq `fpvqq2ď d
2 ÿ
tu,vuPE
pfpuq2`fpvq2q “ d
2dÿ
uPV
fpuq2“
?2dfTf.
The lower bound in (11.2) follows from the following calculation which uses telescopic summation and the ordering of the vertices ofG:
B “ ÿ
tu,vuPE
|fpuq2´fpvq2|
“ ÿ
tu,vuPE,uăv
pfpuq2´fpvq2q
“ ÿ
tu,vuPE,uăv v´1
ÿ
i“u
pfpiq2´fpi`1q2q
“
|V|´1
ÿ
i“1
pfpiq2´fpi`1q2q|Bt1, . . . , iu|
“ ÿ
iPS
pfpiq2´fpi`1q2q|Bt1, . . . , iu|
ě hpGqÿ
iPS
pfpiq2´fpi`1q2qi
“ hpGqÿ
iPS
pfpiqq2
“ hpGqa fTf .
Here we use the fact that|S| ď |V|{2and so|Bt1, . . . , iu|{iěhpGqifiď |V|{2.
Furthermore, notice thatfpi`1q “0fori“ |S|when collapsing the telescopic sum.
11.4.4 Low distortion embeddings of expander graphs
Theorem 11.4.5. Letd ě3 be an integer and let ą 0 be a positive real. For everyd-regular graphG“ pV, Eqandλ2ďd´, we have
c2pGq ě c
2dtlogd|V|u.
In particular, Bourgain’s theorem is tight for families ofd-regular expander graphs.
Proof. For simplicity we assume that|V|is even.
SinceGisd-regular, every vertex hasďdrvertices at distancer. In particular if r “ tlogd|V|u´1, then there are ď |V|{2 vertices at distance r from any given vertex. Define the graphH “ pV, EHqby connecting two vertices if their distance in Gis ě tlogd|V|u. Then the minimal degree of H is ě |V|{2. By a classical theorem of Dirac from 1952 we know that every graph on|V| ě 3 vertices with minimum degree at least|V|{2contains a Hamiltonian cycle; one can find the (simple) proof for instance as Theorem 10.1.1 in the book [1] by Diestel. Since|V|is even, we derive thatH has a perfect matching.
LetB PSV be the adjacency matrix of such a perfect matching. It is a per-mutation matrix of a perper-mutation consisting out of|V|{2disjoint transpositions.
We denote the edges participating in the perfect matching byF. LetAPSV be the adjacency matrix ofG.
Define the matrixY by
Y “dI´A`
2pB´Iq,
and we want to show thatY satisfies the assumptions of Theorem 11.2.1.
It is easy to verify thatY e“0holds. The matrixY is positive semidefinite because for everyxPRV which is perpendicular toewe have the inequality
xTY x “ xTpdI´A`
2pB´Iqqx ě pd´λ2qxTx`
2xTpB´Iqx ě xTx`
2 ÿ
tu,vuPF
p2xpuqxpvq ´xpuq2´xpvq2q ě xTx´
22 ÿ
tu,vuPF
pxpuq2`xpvq2q ě xTx´xTx
ě 0.
To end the proof we only have to evaluateY’s objective value:
´ ÿ
ij:Yijă0
Yijdpxi, xjq2“d|V|
and ÿ
ij:Yiją0
Yijdpxi, xjq2ě
2|V|tlogd|V|u2. Hence,
c2pGq ě c
2dtlogd|V|u.
11.4.5 Construction of a family of expander graphs
Explicit constructions of a family of expander graphs are very much non-trivial.
An easy construction of a family of3-regular expander graphs which neverthe-less relies on a deep result in number theory (Selberg’s 3{16 theorem) is as follows: Letpbe a prime. The vertex set ofGpisZpand a vertexxis connected tox`1,x´1andx´1where all operations are performed modulopand where the inverse of0is defined to be0.