Eigenvalues and expansion of regular graphs - Semantic Scholar
Eigenvalues
NABIL
and Expansion
of Regular
Graphs
KAHALE
Massachusetts
Instituteof
Technology,
Cambridge,
Massachusetts
Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the on the best-known explicit expanders. The spectral method yielded a lower bound of k\4 expansion of Iinear-sized subsets of k-regular Ramanujan graphs. We improve the lower bound k/2, Moreover. we construct afamilyof ontheexpansion of Ramanujan graphs to approximately k-regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k/2. This shows that k/2 is the best bound one can obtain using the second eigenwdue method. We also show an upper bound of roughly induced subgraphs of Ramanujan graphs.
1 + ~ on the average degree of linear-sized This compares positively with the classical bound
2~. As a byproduct, we obtain improved results on random construct selection networks (respectively, extrovert graphs) of smaller than was previously known.
walks on expanders and size (respectively, degree)
Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problems Complexity]: Nonnumerical Algorithms and Problems; G.2.2 [Discrete Mathematics]: Graph Theory General Additional balancing,
Terms: Algorithms,
Theory
Key Words and Phrases: Ramanujan graphs, random
Eigenvalues, expander graphs, walks, selection networks.
Part of this work was done while the author
induced
subgraphs,
load
was at DIMACS.
This work was partially supported by the Defense Advanced Research Projects Agency under Contracts NOO014-92-J-1799 and NOO014-91 -J- 1698, the Air Force under Contract F49620-92-J0125, and the Army under Contract DAAL-03-86-K-0171. This paper was based on “Better
Expansion
for
Ramanujan
grdphs”,
by Nabil
IQdhale, which
ScLence, San Juan, Puerto appeared in the 32nd Annual Symposium on Foundations of Compater Rico, October 1–4, 1991; pp. 398–404. OIEEE, and on “On the Second Eigenvalue and Linear Sympcmum Expansion of Regular Graphs” by Nabil Kahale, which appeared in the 33rd Annaal on Foundations of Computer Science, Pittsburgh, Pennsylvania, October 24–27, 1992: pp. 296–303. Series in Discrete @lEEE. An updated version of the second paper appeared in DIMACS Mathematics and Theoretical Cornpater Science, Volume 10, 1993; pp. 49–62. @American Mathematical Society.
Author’s current CA 94304.
address: XEROX
Palo Aho Research Center, 3333 Coyote Hill
Road, Palo Alto.
Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, the copyright notice, the tide of the publication, and its date appear, and notice is given that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. 01995 ACM 0004-541 1/95/0900-1091 $03.50 Joum~l
of the AsWcldtlon
for Computing
M&chlnery,
Vol
42, No
5, Scptemlxl
IW5, pp
10[)1 – I I [M,
1092
NABIL
KAHALE
1. Introduction Expander graphs are widely used ranging from parallel computation
in Theoretical Computer ] to complexity theory
Science, in areas and cryptography.z
Given an undirected k-regular graph G = (V, E) and a subset X of V, the expansion of X is defined to be the ratio lN(X)l/l Xl, where N(X) = {w ~ V of X, An (a, ~, k, n)-expander is a 3 LI G X, (~1, w) E E} is the set of neighbors k-regular graph on n nodes such that every subset of size at most an has expansion at least f?. It is known that random any ~ < k – 1, there
regular
exists
graphs
a constant
the subsets of a random k-regular least ~. The explicit construction
are good expanders. a such that,
with
In particular,
high
probability,
for all
graph of size at most an have expansion at of expander graphs is much more difficult,
however. The first explicit construction of an infinite family of expanders was discovered by Margulis [1973], and improved in Gabber and Galil [1981], Alon et al. [1987], and Jimbo and Maruoka The best currently known method
[1987]. to calculate
lower
bounds
on the expan-
sion in polynomial time relies on analyzing the second eigenvalue of the graph. Since the adjacency matrix A is symmetric, all its eigenvalues are real and will be denoted by & > Al > ““” > A,l. ~. We have AO = k, and A = max( Al, IA,, _ ~1) < k. Tanner [1984] proved that for any subset X of V, k21Xl IN(X)
I>
(1) AZ + (kz
Therefore, However,
in order to get high for any sequence
– A2)l X1/n
“
expansion, we need A to be as small as possible. G,,, ~ of k-regular graphs on n vertices,
lim inf A(G,,, ~) > 2v’~ as n goes to infinity 1988; Nilli 1991]. Therefore, the best expansion
[Alon 1986: Lubotzky et al. coefficient we can obtain by
applying Tanner’s result is approximately k/4. Ramanujan graphs, which have been explicitly
This bound is achieved by constructed [Lubotzky et al.
1988; Margulis
1988] for many
pairs (k, n). By definition,
a Ramanujan
graph
is
a connected k-regular graph whose eigenvalues + + k are at most 2v”~ in absolute value. The relationship between the eigenvalues of the adjacency matrix and the expansion coefficient has also been investigated in Alon [1986], Alon et al. [1987], Alon and Milman [1985], and Buck [1986], but the bound they get, when applied to nonbipartite Ramanujan graphs and for sufficiently bound. Other results about expanders are large k, is no better than Tanner’s contained in Bien [19891, Lubotzky [to appear]. and Samak [1990]. Some applications, such as the construction of nonblocking networks in for linear-sized Arora et al. [1990], required an expansion greater than k/2 subsets. Indeed, if the expansion of a subset X is greater than k/2, a constant neighbors, that is, neighbors adjacent to only fraction of its nodes have unique one node in X. This allows the construction of a matching between X and N(X) in a logarithmic number of steps and using only local computations. Recently, Pippenger [1993] showed that weak expanders are applications where an expansion greater than k\2 was required.
~ See Ajtai - See Ajtai
et al. [1983], Arora et al. [1990], Pippenger [1993], and Upfal [1989]. et al. [1987], Bellare et al. [1990], Goldreich et al. [1990], and Valiant
sufficient
[ 1976].
in
Eigenl’ahes
and Expansion
of Regular
Graphs
1093
We define the linear expansion of a family of graphs G,, on n vertices to be the best lower bound on the expansion of subsets of size up to an, where a is an arbitrary
small
positive
constant,
sup a>o where
X
ranges
over
if (G. ) is a family bounded
by
the
the
construct
an infinite
inf ~ of
graphs
linear
(1 – ~1 – (4k – 4)/~z of Ramanujan graphs is the other hand, for any modulo 4, and for any
inf n
subsets
of k-regular ~,
that is,
‘~~)’,
G,l of size at most whose
expansion
family
of
k-regular
k/2.
has asymptotically
can
obtain
such k/2
a family is essentially
by the
(G,, )
graphs
and G,l contains
Since
of
an.
largest
We
show
eigenvalue is
at
that
is upper
least
(k/2)
). In particular, the expansion of linear-sized subsets lower bounded by a factor arbitrary close to k/2. On integer k such that k – 1 is a prime congruent to 1 function m of n such that m = o(n), we explicitly
A(G~) s (2 + o(l))v(~ shows that
second
the best lower
second
eigenvalue
construction can be applied values and diameter [Kahale
G.
on
n vertices
such
a subset of size 2m with optimal bound
method.
second
eigenvalue,
on the linear The
that
expansion this
expansion
one
used
this
techniques
in
to prove tightness of relationships between eigen1993]. Clur results provide an efficient way to test
that the expansion of linear-sized subsets of random graphs is at least k/2 – 0(k3/410g]/zk), We also show that the average degree of the induced subgraphs on linear-sized subsets of a k-regular graph G is upper bounded by a factor
arbitra~
2~~). graphs,
This improving
close bound upon
to
1 + ~/2
is equal
+ to
the previous
~z/4
– (k – 1),
1 + ~~
in
known
bound
the
where case
[Alon
~ = max(~, of
Ramanujan
and Chung
1988] of
2JF=i. Sections 3–5 contain our main results. III Section 6, we apply our techniques to obtain improved results on random walks on expanders. Random walks are often used in complexity theory and cryptography, upon previous results in Ajtai et al. [1987] and Applications to selection networks and extrovert Section
7. We conclude
with
some remarks
in Section
and our bound improves Goldreich et al. [1990]. graphs are described in 8.
Some of the results in this paper have appeared in an extended in Kahale [1991; 1993a], and in a more detailed form in Kahale
2. Notation,
Definitions,
and Background
Throughout
the paper,
G = (V, E) will
denote
an undirected
abstract [1993b].
graph
form
on a set V
of vertices. Let L2( V ) denote the set of real-valued functions on V and L~)(v) = {f e L’(v); x ,, ~ ~f(~’) = O}. As usual, we define the scalar product of two vectors f and g of LZ(V) by
and the euclidean norm of a vector f by IIf II = ~. We denote the adjacency matrix of G by A~, or simply by A if there is no risk of confusion. The matrix A is the O-1 n X n matrix whose (i, j) entry is equal to 1 if and only
NABIL
1094 if
(i, ~) = E.
It defines to the vector
~ = L2(V)
a linear operator A~ defined by
(Af)(u)
in
=
L2(V ) that
~
maps
KAHALE
every
vector
(2)
f(w).
((.w)eE This
operator
is selfadjoint
since V~, g e LZ(V),
(Y4f)”g=f”(Ag)
x
=
(3)
f(u)g(w).
(1’,w’)6E For
any matrix
or operator
(i + l)st
largest
by A(G).
For
M
eigenvalue
any subset
with
of M,
real
eigenvalues,
we denote
by A,(G),
and max(Al(G),
Al(A~)
W of V, we denote
by A,(M)
the
IA._ ,(G)I)
by XW the characteristic
vector
of
W xw(~)) = 1, if ~) = W, and O otherwise. We denote the adjacency matrix of the graph induced on W by ALV, the real number A,(AL,, ) by A,(W), and the set of nodes at distance at most 1 from W by Bl(W). For the rest of this section, we assume that G is k-regular. Fact
1988].
2.1 [,S&ang
If
B is a selfadjoint
operator
in a vector
space
L,
then AO(B)
Clearly,
the vector
2.2.
For
g.Bg —
max g= L-{o}
,yJ is an eigenvector
space L~(V) is invariant to L:(V) are AI(G),..., Fact
=
llg112 “ of A with
eigenvalue
under A, and the eigenvalues &_ I(G). Therefore,
any g ● L:(V),
For two column vectors is at most its corresponding
we have
of A
< A1(G)llgll’.
g Ag
g and h., we say that coordinate in h.
k. The vector
of the restriction
g s h if every coordinate
of g
Fact 2.3 [Seneta 1981, page 28, ex. 1.12]. If a real symmetric matrix has only nonnegative entries, its largest eigenvalue is nonnegative and has a corresponding eigenvector with nonnegative components. This eigenvalue is largest in absolute value. Fact is
2.4.
a vector
eigenvalue entries
If a real symmetric with
positive
of B is at most
of B are assumed
Given V’
=
(U,
LI) c
a
H,
graph
V X {O, 1} and
y. This
B has only such
property
still
to be nonnegative
the
where
ccuer ((u,
graph
1), (~’, nz))
that
nonnegative
[Friedman
H’ e
of V’
entries,
and
s
Bs < ys, then the largest holds if only the off-diagonal
H X V’
1991]. is the is an
graph edge
defined if
and
only
on if
E and 1 # m.
Fact 2.5. The negated values. 3. Main
matrix
components
eigenvalues
of H’
consist
of the eigenvalues
of I/
and their
we will
use later
Lemma
In this section, to derive lower
we prove the main lemma bounds on the expansion.
(Theorem
3.6) that
Eigenlalues
and Expansion 3.1.
LEMMA
of Regular
1095
Graphs
If G = (V, E)
is k-regular
on n Lertices,
fAf
Ao(M,~’ ‘)f(l’).
For any L) e X, the value of f on each of the k – Ax( – X is at least f( L’)r(l). Therefore,
PROOF.
of L’ in ~
(Awf)(u) 2 =
(Axf)(u)
+ (k -
(M~’’f)(L)
For
. f > A’llf Ilz, leading 9>0,
L’)
neighbors
Ax(L’))f(u)i’(l)
= ~O(M$’’)f(L).
Since AO(W) < A’, we have (Awf) “fs contradiction that A’ < AO(M,~’ ~), By Claim (Awf)
1097
c!
A’llf Ilz, by Fact 2.1. Assume 3.4 and Claim 3.5, this implies
for that
❑
to a contradiction.
define 1 A4$=Ax+–
(kI-
Ax).
v(~e’ The matrix M$ can be regarded induced on X and has in addition ~x(LI))
on
each node
THEOREM
3.6.
as the matrix of the weighted a loop of weight (k – 1)-1/2
u of X. Suppose
G
=
(V,
E)
is
rnax( AI(G), 2v’~) = 24~cosh 9, where X of V of size at most k-1 \’/VI, we hat,’e A,(M; where the constant PROOF.
from from
Let
behind
1 =
X. A simple Lemma
3.2
) :< i(l
-t
the O is a small
k-regular,
CLAIM PROOF. CLAIM
3.7. This
that
&(W)
s A’,
the mal:rix
> y >0,
follows
we hale
immediately
Since
~’)
sinh((l
absolute
constant.
where
A’ = ~ + 3k1 - ll~z’)
kl~”
is approximated
(x – y)z s 2(coshx from
Taylor’s
= 2~~
shows that 1 = I( 1 + We will use the followby M~,.
– coshy).
expansion
formula.
+
sinh(l(3’) S exp(– < smh((l + 1)6’)
1) f)’)
> (1 +
l)sinh
O’).
6’, we have
sinh(ld’) exp( – O’) – sinh((l —
~ = subset
o(e)),
3.8.
1 —exp(– 1+1 PROOF.
let
[1/2 e] and let W be the set of nodes at distance at most 1 calculation shows that IWI s 3k1 IX I s 3k - ] /(z’ ‘n. It follows
to show that Forx
and
/3 z O. For any nonempty
cosh 0’, with & > 0>0. A straightforward calculation O(6)) and cosh %’ – cosh O = 0(k*f2l/(z’)) = 0(~2). ing inequalities
graph that is exp( – O)(k –
+ 1)0’)
exp(–10’) exp((l
= exp(–(1
– exp(–(1
+ 1)0’) + 1)6’)
– exp(–(1
+ 2)0’) + 1)0’)
sinh %’ sinh((l + 1)6’)
exp(–
0.
Y U N’(Y), ~’
3.9. Note
2.5, we have
Y be the subset
G’ by xl’, with
by Remark by Fact
the By and
= ~!
E’ =
u N’(Y)
u N’(Y))
of
2 e,
we
by f = kxy
we cannot
X
Yin
see
A(1
that
the
+ ~x~fy).
M’f.f
apply
which
s x(]
the
directly may
the
G’ by N’(Y). to
E )).
+ 2klv’1’
X {O}. Denote
3.9,
+ .~(
= “(G)
= A(G),
to
Remark
iS at most
defined
that
set of neighbors applying
+ k— Ivl
AI(G)
of V’ equal
Iwl
Iwpl
< AO(WP) < AI(G)
largest
adjacency
Let
~ =
G,
the
graph
Theorem
be different
the
matrix
of
2v’’=-cosh
eigenvalue
NOW> consider
3.6 from
set of
function
of
O, vertices
the
matrix
f = Lz(y
BY Fact 2.1, + o(6))
llf112.
(7)
The left-hand side is the sum of two terms. The first is equal to A’f. f, and the se~ond corresponds to the weighted self-loops. By eq. (3), we have A’f. f = 2 Ak 2IY 1. On the other hand, since the loops have no weight on Y and have average weight term is equal
– klY]/l exp( – tl)(k – l)-ltz(k to exp(– d)(k – 1)-]/2 (klN’(Y)l
N’(Y)l) on N’(Y), – klYl)~z. Thus,
the second eq. (7) re-
duces to
-,
A’
2ik21Yl
—k(/N’(Y)l
+ v’CTexp(
i(l + 0(~)) (k21Yl + i21N’(Y)l),
< By replacing
IY \ by IX 1,\N’( Y)l by IN(X klXl(k
– 2exp(–6)cosh s lN(X)l(~
Noting that duces to:
simplifications
– 2kexp(–
O)cosh
6)(1
+ O(c)).
(8)
X2 – 2k exp( – O)cosh O =: 2(k exp( 9 ) – 2 cosh 6) cosh 0, eq. 8 re-
[xl
k
the proof
2
using
1 =1– exp( 6 )cosh THEOREM
)1, we get after
0)
IN(X) I
We conclude
– \Yl)
6)
4.2.
If
$
2 exp( O)cosh
O
(1 –
O(E)).
the formula
r
G = (V, E)
l– cosh-O ==
’--
is it-regular
and
“ ~ = max( AI(G),
2-),
then for any nonemp~ subset X of V of size at most k- 1i’ IV 1,the al’erage degree u of the subgraph of G induced on X is at most
(l+i+~(k-l))(l+o(~)) where
the constant
behind
the O is a small
absolute
constant.
1100
NABIL
PROOF.
We use the same notations
as in Theorem
3.6. As noted
KAHALE
before,
the
matrix M~ can be regarded as the matrix of the weighted graph on X that is induced on X and has in addition a loop of weight (k – 1)-1/z exp( – /3)(k – Ax(~I)) on each node u of X. By Fact 2.1, we have xx oM$ Xx < AO(M!)IXI, which
translates
into
k–u s 2~k~(l
o + This
+ O(~))cosh
6,
JCTexp(6)
implies
2(k – l)exp(0)cosh
O– k
CT
11P,,_, g112\ X~,, respectively, We need to show that
PROOF.
and
I/Plz_ ~sllz. Let
Ah = (P + PI,)A(
P + P},).
The
operator
A,,
corresponds
in
some sense to the adjacency matrix of the subgraph induced on B,,(X), but it acts on L,Q(W ). By the conditions of the lemma, there exist positive coefficients a, /3, and y such that P,ZA)lS = yP~s and A,, P,, s = aP,ls + flP,l_, s. By hypothesis, we have A), s < pPs + yPl, s. Premultiplying both sides of this equation by P and Ah yields successively
= /4s – pP~s,
5 p2Ps + p(y
– a)Phs
– ppP,l_ls,
Eigerwalues But
and Expansion
the matrix
entries
A~, I’A~
of Regular
Graphs
– VZP – W( y – a )P}, + p~Pk _ ~ has only
off its diagonal,
and so its largest
2.4). The quadratic form associated definite (Fact 2.1), and so
Since
both
Al,
and
eq. (9) is equal
eigenvalue
to this
P are selfadjoint
llPAhgll
= pllPgll
by hypothesis,
(y-
(Fact
is therefore
semi-
negative
Pz = P, the left-hand
side of
eq. (9) as follows:
a)llPllgll’
- p~llP,z_,g112.
JDI, are
and
(lo)
2 EllP,, _,gll’. selfadjqint,
On the other hand, since P,, AIIS .s, and so allPllsllz
+ pll P},_,sllz = yll Pksll-.
(10) concludes
❑
the proof.
nonnegative
is O since s is positive
and so
a)llp~gll’ A,,
matrix
and since
to IIPA~ g II2. We can rewrite
llPAhg112< p211Pg112 + W(:P But
1101
we have
A,, P,, s .s =
Comparing
this
with
eq.
THEOREM 5.2. For any integer k such that k – 1 is prime, we can explicitly construct an infinite family of k-regular graphs G,, on n Lertices whose linear expansion
is k/2
PROOF.
and such that
We construct
of explicit
Ramanujan
Al(G,,)
the family graphs,
L 2Y’’~(l
+ 2 log ‘log n /log~
(G. ) by altering
so that
the
the known
expansion
n ).
constructions
of (G,, ) is k/2.
From
Lubotzky et al. [1988] and Margulis [1988], we know that we can explicitly construct an infinite family of bipartite Ramanujan graphs (F,, ) on n vertices whose girth c(F~) is (4/3 + o(l)) log~_ ~n. Let F,, = (V, E) be an element of of F,, and 1 = [c(F,, )/21 – 2. Let be k vertices at distance two (u,, vi) = E. The subgraph of F. induced on B ~+,({u}) is a tree not belonging to no cycles. Let u’ and [)’ be two elements the family, L{ e V a vertex neighbors of u and let LI,,
k-regular
graph
. . . . L’k
G,z+ z = (V’,
E’),
where
u,, .,., LL,, be the from LL such that since it contains V. Consider the
V’ = V U {u’, L“}
and
E’ = E
U
U $= ,{(u’,
Ui), (u,, u’), (~)’, ~~,),(~,, ~’)} – U ~=~{(u,, ~,), (L’,, u,)}. Figure 1 shows the graph G.+ ~ in the neighborhood of u in the case k = 3. For shorthand, we AF,, W A> ‘G,,+, denote by A’ and AI(A’ ) by A’. We need to show that A’ < (2 + o(l))~~.
Assume
and let A’ = 2~k~cosh corresponding to A’.
that
0’, with
A’ > 2v’~ (3’ > 0. Let
(otherwise, g 6 L~,(V’)
we are done),
be an eigenvector
We outline informally the basic ideas of the proof. Roughly speaking, we will show that the values that g takes on the nodes u’, L’, u,, 1~1 are small compared to Ilg 11.This implies that g is close in /z-norm to its r~estriction f on V. Lemma But since g(u’), g(~i’~, 3.1 then implies that f “Af < (2 + o(l))~~llgll-. g(ui),
and g(L’i)
are small,
the scalar
product
f ~Af
neighbors
in
is close
to g . A’g
=
A’ Ilgll-,
and so A’ < (2 + o(l))~~”. Since g(u)
u and
= g(u’).
LL’ have
the
same
G,,. ~ and
A’ # O, we
have
By eq. (3), we have
A’1/g112 == g.A’g
=fAf
-2
fig ~=1
+ 2 fg(L1’)g(Lt,) 1=1
+ 2
&L’’)@.
[=1
(11)
1102
NABIL
u.
KAHALE
.U( ,, ,},.,, ., ‘, ~
,’
u ~ ‘,
U1 # ,, FIG. 1. The graph G,l, * in the neighborhood in the case k = 3. The belonging to E – E’.
dotted
edges
are
of u those
‘:/./’;,” / *
JV2
–2g(~i)g(~i) by g(ui)2 = A’g(u’) implies that
A similar
holds
for
/illg112 sf”flf+
~’. Combining
f“flfs
1 + ;
A,(A) llf112 + :(g(u’) s 2J=(llg112
this with
()
We use Lemma 3.1 to bound the term – g(u’) since g E L:( V’), and so
+ g(~,)2.
; i=l
[g(u,
~” A~. Note
&
“ \
;\t\
\,
On
/“,
,\\ji,l
i,!J;’’,l,!\li~\A\
We upper bound equality (A’g)(u’)
, ,./’”””
:~~
,\’, !;,, ,, ,,
relation
‘~~ 3 .,,*V ,,-..
the
;’i l,,
other
\
hand,
the
eq. (11), we get
)2 +g(L’, that
)2)
1/2
cosh~(l(l’)x~=,
~g(u,)z. ial g(ul)z.
Combining
this
with
eq. (13),
we
get
8 (14)
cosh 6’ < 1 + coshzlo’ Solving
eq. (14) yields
Theorem
for sufficiently
large
n, and so
L ‘7)(1
A’=2JF71+;
k/2.
/3’ s (log 1)/1
“
logs log n
O+ O(1)) 7. This shows that Ramanujan graphs of degree at least 7 are extrovert graphs. Classical results [Alon and Chung
1989] require
8. Concluding (1) Let
H
the degree
Remarks
be a graph
and Further of maximum
to be at least
15.
Work degree
at most
k, and ~ a real number
no
smaller than 2~~. Theorem 3.6 implies that, if there exists an infinite family G. of k-regular graphs ccmtaining H as an induced subgraph and such that A(G,, ) < (1 + o(l))~, then Atl(lffi) s j. (M: can be defined congruent to 1 modulo 4, this similarly to M:.) If k – 1 is al prime condition
can be shown
to be sufficient
(2) It is still an open question graphs with linear expansion
whether at most
[Kahale there k/2.
1993b].
exists
a family
of Ramanujan
(3) It would be interesting to calculate the exact value of the linear expansion of the known explicit constructions of Ramanujan graphs [Lubotzky et al. 1988; Margulis 1988]. Theorem 4.1 shows that it is at least k/2. On the other hand, an easy combinatorial argument shows that the linear expansion of any family of k-regular graphs is at most k – 1. Besides being Ramanujan, the graphs constructed in Lubotzky et al. [1988] and Margulis [1988] have other interesting combinatorial properties. For example, they are Cayley graphs and have high girth, unlike the graphs that we constructed in Section 5. This leads us to conjecture that their linear expansion is strictly greater than k/2. Any explicit construction of k-regular graphs with
provable
linear
expansion
strictly
greater
k/2
than
would
also
be
interesting. REFERENCES AJTAI.
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] 995: ACCEPTED NIAY 1995
Mach! nery. Vo[
42. N
NABIL
and Expansion
of Regular
Graphs
KAHALE
Massachusetts
Instituteof
Technology,
Cambridge,
Massachusetts
Abstract. The spectral method is the best currently known technique to prove lower bounds on expansion. Ramanujan graphs, which have asymptotically optimal second eigenvalue, are the on the best-known explicit expanders. The spectral method yielded a lower bound of k\4 expansion of Iinear-sized subsets of k-regular Ramanujan graphs. We improve the lower bound k/2, Moreover. we construct afamilyof ontheexpansion of Ramanujan graphs to approximately k-regular graphs with asymptotically optimal second eigenvalue and linear expansion equal to k/2. This shows that k/2 is the best bound one can obtain using the second eigenwdue method. We also show an upper bound of roughly induced subgraphs of Ramanujan graphs.
1 + ~ on the average degree of linear-sized This compares positively with the classical bound
2~. As a byproduct, we obtain improved results on random construct selection networks (respectively, extrovert graphs) of smaller than was previously known.
walks on expanders and size (respectively, degree)
Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problems Complexity]: Nonnumerical Algorithms and Problems; G.2.2 [Discrete Mathematics]: Graph Theory General Additional balancing,
Terms: Algorithms,
Theory
Key Words and Phrases: Ramanujan graphs, random
Eigenvalues, expander graphs, walks, selection networks.
Part of this work was done while the author
induced
subgraphs,
load
was at DIMACS.
This work was partially supported by the Defense Advanced Research Projects Agency under Contracts NOO014-92-J-1799 and NOO014-91 -J- 1698, the Air Force under Contract F49620-92-J0125, and the Army under Contract DAAL-03-86-K-0171. This paper was based on “Better
Expansion
for
Ramanujan
grdphs”,
by Nabil
IQdhale, which
ScLence, San Juan, Puerto appeared in the 32nd Annual Symposium on Foundations of Compater Rico, October 1–4, 1991; pp. 398–404. OIEEE, and on “On the Second Eigenvalue and Linear Sympcmum Expansion of Regular Graphs” by Nabil Kahale, which appeared in the 33rd Annaal on Foundations of Computer Science, Pittsburgh, Pennsylvania, October 24–27, 1992: pp. 296–303. Series in Discrete @lEEE. An updated version of the second paper appeared in DIMACS Mathematics and Theoretical Cornpater Science, Volume 10, 1993; pp. 49–62. @American Mathematical Society.
Author’s current CA 94304.
address: XEROX
Palo Aho Research Center, 3333 Coyote Hill
Road, Palo Alto.
Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage, the copyright notice, the tide of the publication, and its date appear, and notice is given that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. 01995 ACM 0004-541 1/95/0900-1091 $03.50 Joum~l
of the AsWcldtlon
for Computing
M&chlnery,
Vol
42, No
5, Scptemlxl
IW5, pp
10[)1 – I I [M,
1092
NABIL
KAHALE
1. Introduction Expander graphs are widely used ranging from parallel computation
in Theoretical Computer ] to complexity theory
Science, in areas and cryptography.z
Given an undirected k-regular graph G = (V, E) and a subset X of V, the expansion of X is defined to be the ratio lN(X)l/l Xl, where N(X) = {w ~ V of X, An (a, ~, k, n)-expander is a 3 LI G X, (~1, w) E E} is the set of neighbors k-regular graph on n nodes such that every subset of size at most an has expansion at least f?. It is known that random any ~ < k – 1, there
regular
exists
graphs
a constant
the subsets of a random k-regular least ~. The explicit construction
are good expanders. a such that,
with
In particular,
high
probability,
for all
graph of size at most an have expansion at of expander graphs is much more difficult,
however. The first explicit construction of an infinite family of expanders was discovered by Margulis [1973], and improved in Gabber and Galil [1981], Alon et al. [1987], and Jimbo and Maruoka The best currently known method
[1987]. to calculate
lower
bounds
on the expan-
sion in polynomial time relies on analyzing the second eigenvalue of the graph. Since the adjacency matrix A is symmetric, all its eigenvalues are real and will be denoted by & > Al > ““” > A,l. ~. We have AO = k, and A = max( Al, IA,, _ ~1) < k. Tanner [1984] proved that for any subset X of V, k21Xl IN(X)
I>
(1) AZ + (kz
Therefore, However,
in order to get high for any sequence
– A2)l X1/n
“
expansion, we need A to be as small as possible. G,,, ~ of k-regular graphs on n vertices,
lim inf A(G,,, ~) > 2v’~ as n goes to infinity 1988; Nilli 1991]. Therefore, the best expansion
[Alon 1986: Lubotzky et al. coefficient we can obtain by
applying Tanner’s result is approximately k/4. Ramanujan graphs, which have been explicitly
This bound is achieved by constructed [Lubotzky et al.
1988; Margulis
1988] for many
pairs (k, n). By definition,
a Ramanujan
graph
is
a connected k-regular graph whose eigenvalues + + k are at most 2v”~ in absolute value. The relationship between the eigenvalues of the adjacency matrix and the expansion coefficient has also been investigated in Alon [1986], Alon et al. [1987], Alon and Milman [1985], and Buck [1986], but the bound they get, when applied to nonbipartite Ramanujan graphs and for sufficiently bound. Other results about expanders are large k, is no better than Tanner’s contained in Bien [19891, Lubotzky [to appear]. and Samak [1990]. Some applications, such as the construction of nonblocking networks in for linear-sized Arora et al. [1990], required an expansion greater than k/2 subsets. Indeed, if the expansion of a subset X is greater than k/2, a constant neighbors, that is, neighbors adjacent to only fraction of its nodes have unique one node in X. This allows the construction of a matching between X and N(X) in a logarithmic number of steps and using only local computations. Recently, Pippenger [1993] showed that weak expanders are applications where an expansion greater than k\2 was required.
~ See Ajtai - See Ajtai
et al. [1983], Arora et al. [1990], Pippenger [1993], and Upfal [1989]. et al. [1987], Bellare et al. [1990], Goldreich et al. [1990], and Valiant
sufficient
[ 1976].
in
Eigenl’ahes
and Expansion
of Regular
Graphs
1093
We define the linear expansion of a family of graphs G,, on n vertices to be the best lower bound on the expansion of subsets of size up to an, where a is an arbitrary
small
positive
constant,
sup a>o where
X
ranges
over
if (G. ) is a family bounded
by
the
the
construct
an infinite
inf ~ of
graphs
linear
(1 – ~1 – (4k – 4)/~z of Ramanujan graphs is the other hand, for any modulo 4, and for any
inf n
subsets
of k-regular ~,
that is,
‘~~)’,
G,l of size at most whose
expansion
family
of
k-regular
k/2.
has asymptotically
can
obtain
such k/2
a family is essentially
by the
(G,, )
graphs
and G,l contains
Since
of
an.
largest
We
show
eigenvalue is
at
that
is upper
least
(k/2)
). In particular, the expansion of linear-sized subsets lower bounded by a factor arbitrary close to k/2. On integer k such that k – 1 is a prime congruent to 1 function m of n such that m = o(n), we explicitly
A(G~) s (2 + o(l))v(~ shows that
second
the best lower
second
eigenvalue
construction can be applied values and diameter [Kahale
G.
on
n vertices
such
a subset of size 2m with optimal bound
method.
second
eigenvalue,
on the linear The
that
expansion this
expansion
one
used
this
techniques
in
to prove tightness of relationships between eigen1993]. Clur results provide an efficient way to test
that the expansion of linear-sized subsets of random graphs is at least k/2 – 0(k3/410g]/zk), We also show that the average degree of the induced subgraphs on linear-sized subsets of a k-regular graph G is upper bounded by a factor
arbitra~
2~~). graphs,
This improving
close bound upon
to
1 + ~/2
is equal
+ to
the previous
~z/4
– (k – 1),
1 + ~~
in
known
bound
the
where case
[Alon
~ = max(~, of
Ramanujan
and Chung
1988] of
2JF=i. Sections 3–5 contain our main results. III Section 6, we apply our techniques to obtain improved results on random walks on expanders. Random walks are often used in complexity theory and cryptography, upon previous results in Ajtai et al. [1987] and Applications to selection networks and extrovert Section
7. We conclude
with
some remarks
in Section
and our bound improves Goldreich et al. [1990]. graphs are described in 8.
Some of the results in this paper have appeared in an extended in Kahale [1991; 1993a], and in a more detailed form in Kahale
2. Notation,
Definitions,
and Background
Throughout
the paper,
G = (V, E) will
denote
an undirected
abstract [1993b].
graph
form
on a set V
of vertices. Let L2( V ) denote the set of real-valued functions on V and L~)(v) = {f e L’(v); x ,, ~ ~f(~’) = O}. As usual, we define the scalar product of two vectors f and g of LZ(V) by
and the euclidean norm of a vector f by IIf II = ~. We denote the adjacency matrix of G by A~, or simply by A if there is no risk of confusion. The matrix A is the O-1 n X n matrix whose (i, j) entry is equal to 1 if and only
NABIL
1094 if
(i, ~) = E.
It defines to the vector
~ = L2(V)
a linear operator A~ defined by
(Af)(u)
in
=
L2(V ) that
~
maps
KAHALE
every
vector
(2)
f(w).
((.w)eE This
operator
is selfadjoint
since V~, g e LZ(V),
(Y4f)”g=f”(Ag)
x
=
(3)
f(u)g(w).
(1’,w’)6E For
any matrix
or operator
(i + l)st
largest
by A(G).
For
M
eigenvalue
any subset
with
of M,
real
eigenvalues,
we denote
by A,(G),
and max(Al(G),
Al(A~)
W of V, we denote
by A,(M)
the
IA._ ,(G)I)
by XW the characteristic
vector
of
W xw(~)) = 1, if ~) = W, and O otherwise. We denote the adjacency matrix of the graph induced on W by ALV, the real number A,(AL,, ) by A,(W), and the set of nodes at distance at most 1 from W by Bl(W). For the rest of this section, we assume that G is k-regular. Fact
1988].
2.1 [,S&ang
If
B is a selfadjoint
operator
in a vector
space
L,
then AO(B)
Clearly,
the vector
2.2.
For
g.Bg —
max g= L-{o}
,yJ is an eigenvector
space L~(V) is invariant to L:(V) are AI(G),..., Fact
=
llg112 “ of A with
eigenvalue
under A, and the eigenvalues &_ I(G). Therefore,
any g ● L:(V),
For two column vectors is at most its corresponding
we have
of A
< A1(G)llgll’.
g Ag
g and h., we say that coordinate in h.
k. The vector
of the restriction
g s h if every coordinate
of g
Fact 2.3 [Seneta 1981, page 28, ex. 1.12]. If a real symmetric matrix has only nonnegative entries, its largest eigenvalue is nonnegative and has a corresponding eigenvector with nonnegative components. This eigenvalue is largest in absolute value. Fact is
2.4.
a vector
eigenvalue entries
If a real symmetric with
positive
of B is at most
of B are assumed
Given V’
=
(U,
LI) c
a
H,
graph
V X {O, 1} and
y. This
B has only such
property
still
to be nonnegative
the
where
ccuer ((u,
graph
1), (~’, nz))
that
nonnegative
[Friedman
H’ e
of V’
entries,
and
s
Bs < ys, then the largest holds if only the off-diagonal
H X V’
1991]. is the is an
graph edge
defined if
and
only
on if
E and 1 # m.
Fact 2.5. The negated values. 3. Main
matrix
components
eigenvalues
of H’
consist
of the eigenvalues
of I/
and their
we will
use later
Lemma
In this section, to derive lower
we prove the main lemma bounds on the expansion.
(Theorem
3.6) that
Eigenlalues
and Expansion 3.1.
LEMMA
of Regular
1095
Graphs
If G = (V, E)
is k-regular
on n Lertices,
fAf
Ao(M,~’ ‘)f(l’).
For any L) e X, the value of f on each of the k – Ax( – X is at least f( L’)r(l). Therefore,
PROOF.
of L’ in ~
(Awf)(u) 2 =
(Axf)(u)
+ (k -
(M~’’f)(L)
For
. f > A’llf Ilz, leading 9>0,
L’)
neighbors
Ax(L’))f(u)i’(l)
= ~O(M$’’)f(L).
Since AO(W) < A’, we have (Awf) “fs contradiction that A’ < AO(M,~’ ~), By Claim (Awf)
1097
c!
A’llf Ilz, by Fact 2.1. Assume 3.4 and Claim 3.5, this implies
for that
❑
to a contradiction.
define 1 A4$=Ax+–
(kI-
Ax).
v(~e’ The matrix M$ can be regarded induced on X and has in addition ~x(LI))
on
each node
THEOREM
3.6.
as the matrix of the weighted a loop of weight (k – 1)-1/2
u of X. Suppose
G
=
(V,
E)
is
rnax( AI(G), 2v’~) = 24~cosh 9, where X of V of size at most k-1 \’/VI, we hat,’e A,(M; where the constant PROOF.
from from
Let
behind
1 =
X. A simple Lemma
3.2
) :< i(l
-t
the O is a small
k-regular,
CLAIM PROOF. CLAIM
3.7. This
that
&(W)
s A’,
the mal:rix
> y >0,
follows
we hale
immediately
Since
~’)
sinh((l
absolute
constant.
where
A’ = ~ + 3k1 - ll~z’)
kl~”
is approximated
(x – y)z s 2(coshx from
Taylor’s
= 2~~
shows that 1 = I( 1 + We will use the followby M~,.
– coshy).
expansion
formula.
+
sinh(l(3’) S exp(– < smh((l + 1)6’)
1) f)’)
> (1 +
l)sinh
O’).
6’, we have
sinh(ld’) exp( – O’) – sinh((l —
~ = subset
o(e)),
3.8.
1 —exp(– 1+1 PROOF.
let
[1/2 e] and let W be the set of nodes at distance at most 1 calculation shows that IWI s 3k1 IX I s 3k - ] /(z’ ‘n. It follows
to show that Forx
and
/3 z O. For any nonempty
cosh 0’, with & > 0>0. A straightforward calculation O(6)) and cosh %’ – cosh O = 0(k*f2l/(z’)) = 0(~2). ing inequalities
graph that is exp( – O)(k –
+ 1)0’)
exp(–10’) exp((l
= exp(–(1
– exp(–(1
+ 1)0’) + 1)6’)
– exp(–(1
+ 2)0’) + 1)0’)
sinh %’ sinh((l + 1)6’)
exp(–
0.
Y U N’(Y), ~’
3.9. Note
2.5, we have
Y be the subset
G’ by xl’, with
by Remark by Fact
the By and
= ~!
E’ =
u N’(Y)
u N’(Y))
of
2 e,
we
by f = kxy
we cannot
X
Yin
see
A(1
that
the
+ ~x~fy).
M’f.f
apply
which
s x(]
the
directly may
the
G’ by N’(Y). to
E )).
+ 2klv’1’
X {O}. Denote
3.9,
+ .~(
= “(G)
= A(G),
to
Remark
iS at most
defined
that
set of neighbors applying
+ k— Ivl
AI(G)
of V’ equal
Iwl
Iwpl
< AO(WP) < AI(G)
largest
adjacency
Let
~ =
G,
the
graph
Theorem
be different
the
matrix
of
2v’’=-cosh
eigenvalue
NOW> consider
3.6 from
set of
function
of
O, vertices
the
matrix
f = Lz(y
BY Fact 2.1, + o(6))
llf112.
(7)
The left-hand side is the sum of two terms. The first is equal to A’f. f, and the se~ond corresponds to the weighted self-loops. By eq. (3), we have A’f. f = 2 Ak 2IY 1. On the other hand, since the loops have no weight on Y and have average weight term is equal
– klY]/l exp( – tl)(k – l)-ltz(k to exp(– d)(k – 1)-]/2 (klN’(Y)l
N’(Y)l) on N’(Y), – klYl)~z. Thus,
the second eq. (7) re-
duces to
-,
A’
2ik21Yl
—k(/N’(Y)l
+ v’CTexp(
i(l + 0(~)) (k21Yl + i21N’(Y)l),
< By replacing
IY \ by IX 1,\N’( Y)l by IN(X klXl(k
– 2exp(–6)cosh s lN(X)l(~
Noting that duces to:
simplifications
– 2kexp(–
O)cosh
6)(1
+ O(c)).
(8)
X2 – 2k exp( – O)cosh O =: 2(k exp( 9 ) – 2 cosh 6) cosh 0, eq. 8 re-
[xl
k
the proof
2
using
1 =1– exp( 6 )cosh THEOREM
)1, we get after
0)
IN(X) I
We conclude
– \Yl)
6)
4.2.
If
$
2 exp( O)cosh
O
(1 –
O(E)).
the formula
r
G = (V, E)
l– cosh-O ==
’--
is it-regular
and
“ ~ = max( AI(G),
2-),
then for any nonemp~ subset X of V of size at most k- 1i’ IV 1,the al’erage degree u of the subgraph of G induced on X is at most
(l+i+~(k-l))(l+o(~)) where
the constant
behind
the O is a small
absolute
constant.
1100
NABIL
PROOF.
We use the same notations
as in Theorem
3.6. As noted
KAHALE
before,
the
matrix M~ can be regarded as the matrix of the weighted graph on X that is induced on X and has in addition a loop of weight (k – 1)-1/z exp( – /3)(k – Ax(~I)) on each node u of X. By Fact 2.1, we have xx oM$ Xx < AO(M!)IXI, which
translates
into
k–u s 2~k~(l
o + This
+ O(~))cosh
6,
JCTexp(6)
implies
2(k – l)exp(0)cosh
O– k
CT
11P,,_, g112\ X~,, respectively, We need to show that
PROOF.
and
I/Plz_ ~sllz. Let
Ah = (P + PI,)A(
P + P},).
The
operator
A,,
corresponds
in
some sense to the adjacency matrix of the subgraph induced on B,,(X), but it acts on L,Q(W ). By the conditions of the lemma, there exist positive coefficients a, /3, and y such that P,ZA)lS = yP~s and A,, P,, s = aP,ls + flP,l_, s. By hypothesis, we have A), s < pPs + yPl, s. Premultiplying both sides of this equation by P and Ah yields successively
= /4s – pP~s,
5 p2Ps + p(y
– a)Phs
– ppP,l_ls,
Eigerwalues But
and Expansion
the matrix
entries
A~, I’A~
of Regular
Graphs
– VZP – W( y – a )P}, + p~Pk _ ~ has only
off its diagonal,
and so its largest
2.4). The quadratic form associated definite (Fact 2.1), and so
Since
both
Al,
and
eq. (9) is equal
eigenvalue
to this
P are selfadjoint
llPAhgll
= pllPgll
by hypothesis,
(y-
(Fact
is therefore
semi-
negative
Pz = P, the left-hand
side of
eq. (9) as follows:
a)llPllgll’
- p~llP,z_,g112.
JDI, are
and
(lo)
2 EllP,, _,gll’. selfadjqint,
On the other hand, since P,, AIIS .s, and so allPllsllz
+ pll P},_,sllz = yll Pksll-.
(10) concludes
❑
the proof.
nonnegative
is O since s is positive
and so
a)llp~gll’ A,,
matrix
and since
to IIPA~ g II2. We can rewrite
llPAhg112< p211Pg112 + W(:P But
1101
we have
A,, P,, s .s =
Comparing
this
with
eq.
THEOREM 5.2. For any integer k such that k – 1 is prime, we can explicitly construct an infinite family of k-regular graphs G,, on n Lertices whose linear expansion
is k/2
PROOF.
and such that
We construct
of explicit
Ramanujan
Al(G,,)
the family graphs,
L 2Y’’~(l
+ 2 log ‘log n /log~
(G. ) by altering
so that
the
the known
expansion
n ).
constructions
of (G,, ) is k/2.
From
Lubotzky et al. [1988] and Margulis [1988], we know that we can explicitly construct an infinite family of bipartite Ramanujan graphs (F,, ) on n vertices whose girth c(F~) is (4/3 + o(l)) log~_ ~n. Let F,, = (V, E) be an element of of F,, and 1 = [c(F,, )/21 – 2. Let be k vertices at distance two (u,, vi) = E. The subgraph of F. induced on B ~+,({u}) is a tree not belonging to no cycles. Let u’ and [)’ be two elements the family, L{ e V a vertex neighbors of u and let LI,,
k-regular
graph
. . . . L’k
G,z+ z = (V’,
E’),
where
u,, .,., LL,, be the from LL such that since it contains V. Consider the
V’ = V U {u’, L“}
and
E’ = E
U
U $= ,{(u’,
Ui), (u,, u’), (~)’, ~~,),(~,, ~’)} – U ~=~{(u,, ~,), (L’,, u,)}. Figure 1 shows the graph G.+ ~ in the neighborhood of u in the case k = 3. For shorthand, we AF,, W A> ‘G,,+, denote by A’ and AI(A’ ) by A’. We need to show that A’ < (2 + o(l))~~.
Assume
and let A’ = 2~k~cosh corresponding to A’.
that
0’, with
A’ > 2v’~ (3’ > 0. Let
(otherwise, g 6 L~,(V’)
we are done),
be an eigenvector
We outline informally the basic ideas of the proof. Roughly speaking, we will show that the values that g takes on the nodes u’, L’, u,, 1~1 are small compared to Ilg 11.This implies that g is close in /z-norm to its r~estriction f on V. Lemma But since g(u’), g(~i’~, 3.1 then implies that f “Af < (2 + o(l))~~llgll-. g(ui),
and g(L’i)
are small,
the scalar
product
f ~Af
neighbors
in
is close
to g . A’g
=
A’ Ilgll-,
and so A’ < (2 + o(l))~~”. Since g(u)
u and
= g(u’).
LL’ have
the
same
G,,. ~ and
A’ # O, we
have
By eq. (3), we have
A’1/g112 == g.A’g
=fAf
-2
fig ~=1
+ 2 fg(L1’)g(Lt,) 1=1
+ 2
&L’’)@.
[=1
(11)
1102
NABIL
u.
KAHALE
.U( ,, ,},.,, ., ‘, ~
,’
u ~ ‘,
U1 # ,, FIG. 1. The graph G,l, * in the neighborhood in the case k = 3. The belonging to E – E’.
dotted
edges
are
of u those
‘:/./’;,” / *
JV2
–2g(~i)g(~i) by g(ui)2 = A’g(u’) implies that
A similar
holds
for
/illg112 sf”flf+
~’. Combining
f“flfs
1 + ;
A,(A) llf112 + :(g(u’) s 2J=(llg112
this with
()
We use Lemma 3.1 to bound the term – g(u’) since g E L:( V’), and so
+ g(~,)2.
; i=l
[g(u,
~” A~. Note
&
“ \
;\t\
\,
On
/“,
,\\ji,l
i,!J;’’,l,!\li~\A\
We upper bound equality (A’g)(u’)
, ,./’”””
:~~
,\’, !;,, ,, ,,
relation
‘~~ 3 .,,*V ,,-..
the
;’i l,,
other
\
hand,
the
eq. (11), we get
)2 +g(L’, that
)2)
1/2
cosh~(l(l’)x~=,
~g(u,)z. ial g(ul)z.
Combining
this
with
eq. (13),
we
get
8 (14)
cosh 6’ < 1 + coshzlo’ Solving
eq. (14) yields
Theorem
for sufficiently
large
n, and so
L ‘7)(1
A’=2JF71+;
k/2.
/3’ s (log 1)/1
“
logs log n
O+ O(1)) 7. This shows that Ramanujan graphs of degree at least 7 are extrovert graphs. Classical results [Alon and Chung
1989] require
8. Concluding (1) Let
H
the degree
Remarks
be a graph
and Further of maximum
to be at least
15.
Work degree
at most
k, and ~ a real number
no
smaller than 2~~. Theorem 3.6 implies that, if there exists an infinite family G. of k-regular graphs ccmtaining H as an induced subgraph and such that A(G,, ) < (1 + o(l))~, then Atl(lffi) s j. (M: can be defined congruent to 1 modulo 4, this similarly to M:.) If k – 1 is al prime condition
can be shown
to be sufficient
(2) It is still an open question graphs with linear expansion
whether at most
[Kahale there k/2.
1993b].
exists
a family
of Ramanujan
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