An infinity of Ramanujan graphs

Introduction and setupPrerequisites

Proof of the main theorem

An infinity of Ramanujan graphsAfter Adam Marcus, Daniel Spielman, and Nikhil Srivastava

Andreas Næs Aaserud

Department of Mathematics,UCLA

4 November 2013

Andreas Næs Aaserud An infinity of Ramanujan graphs



Outline

1 Introduction and setup

2 Prerequisites

3 Proof of the main theorem




We will prove the following breakthrough result of the paperInterlacing Families I: Bipartite Ramanujan graphs of all degrees:

Theorem (Marcus-Spielman-Srivastava ’13)

There exists an infinite family of d-regular Ramanujan graphs forevery d ≥ 3.

Note: Prior to this paper, the existence of such a family was onlyknown for d of the form 1 + q, where q is a prime power.

Definition

Let G be a graph with adjacency matrix A. If G is d-regular and

σ(G ) := σ(A) ⊆ {±d} ∪ [−2√d − 1, 2

√d − 1],

then we call G a Ramanujan graph.

Note: All d-regular graphs G satisfy d ∈ σ(G ).Andreas Næs Aaserud An infinity of Ramanujan graphs



Moreover, if G is bipartite then σ(G ) = −σ(G ). (Indeed, theadjacency matrix of such a graph is of the form(

0 AAT 0

).

Thus if (x , y)T is an eigenvector with eigenvalue λ, then (x ,−y)T

is an eigenvector with eigenvalue −λ.)

Note: The range of eigenvalues in the definition of a Ramanujangraph cannot be shrunk if one hopes to construct an infinite familyof such graphs. Indeed it was proved by Alon and Boppana in themid-80s that for every ε > 0, every sufficiently large d-regulargraph has an eigenvalue λ such that 2

√d − 1− ε ≤ |λ| < d .




2-lifts and signings

Definition

Let G = (V ,E ) be a graph. A 2-lift of G is a graph

1. that has two vertices f (v) = {v0, v1} (called the fiber over v)for each vertex v of V , and

2. that, for each edge (u, v) ∈ E with f (v) = {v0, v1} andf (u) = {u0, u1}, contains exactly one of the following twopairs of edges:

{(u0, v0), (u1, v1)} or {(u0, v1), (u1, v0)} (*)

Note: Given a graph G , the 2-lifts of G are in 1–1 correspondencewith so-called signings of (the edges of) G , i.e., maps E → {±1}:The sign of an edge corresponds to the choice (*).




A lemma of Bilu and Linial

We associate to each signing s the signed adjacency matrix As .

Lemma (Bilu-Linial ’06)

Let s be a signing of a graph G = (V ,E ). The spectrum of the2-lift corresponding to s is the union of those of A and As ,counting multiplicities.

Proof. The adjacency matrix of the 2-lift G corresponding to s isof the form

A =

(A1 A2

A2 A1

),

where A1 is the adjacency matrix of (V , s−1(1)) and A2 is theadjacency matrix of (V , s−1(−1)). (Just order the vertices of G asv1

1 , . . . , v1n , v

21 , . . . , v

2n , where {v1

j , v2j } is the fiber over vj ∈ V .)




A lemma of Bilu and Linial, cont.

Moreover, A = A1 + A2 and As = A1 − A2, because s−1(1) ands−1(−1) are disjoint sets of edges. Now, if v is an eigenvector forA corresponding to an eigenvalue λ, then

A(v , v)T = (Av ,Av)T = λ(v , v)T

and, if v is an eigenvector for As corresponding to an eigenvalue µ,then

A(v ,−v)T = (Asv ,−Asv)T = µ(v ,−v)T .

By dimension considerations, we get in this way all of theeigenvalues of A with the desired multiplicities. (Note that we aredealing with diagonalizable matrices!) �




Matching polynomials

Given a graph G , denote by mi the number of matchings in G withi edges, i.e., collections of i non-adjacent edges in G . Denote thenumber of vertices by n.

Definition (Heilmann-Lieb ’72)

The matching polynomial µG is the polynomial

µG (x) =∑i≥0

(−1)imixn−2i .

The authors use the following theorem, whose proof we skip:

Theorem (Heilmann-Lieb ’72)

Let G be a graph. Then µG (x) has only real roots. Moreover, if Ghas maximum degree d , then all roots of µG (x) have absolutevalue at most 2

√d − 1.




Overview

Let G be a d-regular bipartite Ramanujan graph with adjacencymatrix A.

1◦ We will first prove that

Es [fs(x)] = µG (x),

where fs(x) is the characteristic polynomial of As and s variesthrough all possible signings of G .

2◦ We next prove that {fs(x)}s is an interlacing family. It willfollow that there exists a signing s such that every root offs(x) is at most the largest root of Es [fs(x)].

3◦ By the result of Heilmann-Lieb and step 1◦, there exists a2-lift of G that is also a d-regular bipartite Ramanujan graph.

Finally, if we take appropriate successive 2-lifts of a completed-regular bipartite graph, we get an infinite family of d-regular(bipartite) Ramanujan graphs.




Step 1: Expected characteristic polynomial

We are given a finite graph G = (V ,E ) and number the vertices1, . . . , n and the edges 1, . . . ,m. We also identify edges with pairs(i , j) of vertices.

Theorem

Es∈{±1}m [fs(x)] = µG (x).

Proof. The theorem will follow from three simple observations:1◦. The number of k-matchings (i.e., matchings with k edges)equals the number of permutations of the vertices that areproducts of k disjoint 2-cycles, where each 2-cycle corresponds toan edge.




Step 1: Expected characteristic polynomial, cont.

2◦. In the expression

Es∈{±1}m [fs(x)] =∑σ∈Σn

sgn(σ)Es∈{±1}m

[n∏

i=1

(xI − As)i ,σ(i)

],

we need only sum over products of disjoint 2-cycles.(Indeed, let σ ∈ Σn contain a cycle of length at least 3, saymapping i1 to i2 and i2 to i3. Then the product in the expressionabove contains the factors si1,i2 and si2,i3 exactly once.Independence implies

Es∈{±1}m

[n∏

i=1

(xI − As)i ,σ(i)

]= 0.)




Step 1: Expected characteristic polynomial, cont.

3◦. Given k ∈ N and a permutation σ that is the product of kdisjoint 2-cycles, we have, independently of s ∈ {±1}m, that

∏i

(As)i ,σ(i) =

{1 if all of the 2-cycles correspond to edges

0 otherwise,

where the product runs over all i that are not fixed by σ.Finally, with mk = #{k-matchings}, these observations imply that

Es∈{±1}m [fs(x)] =∑k

(−1)kxn−2k∑σ

Es∈{±1}m

[∏i

(As)i ,σ(i)

]=∑k

(−1)kxn−2kmk = µG (x). �




Method of interlacing polynomials

We will now describe the general technique that allowed theauthors to prove their result.

Definition

We say that a polynomial g(x) = a∏n−1

i=1 (x − αi ) interlaces apolynomial f (x) = b

∏ni=1(x − βi ) if αi , βi , a, b ∈ R and

β1 ≤ α1 ≤ β2 ≤ · · · ≤ αn−1 ≤ βn.

Lemma

Let f1, . . . , fk ∈ R[x ] be polynomials of the same degree withpositive leading coefficients. If they have a common interlacing,then there exists an i such that the largest root of fi is at most thelargest (real) root of

∑kj=1 fj .




Method of interlacing polynomials, cont.

Proof of the lemma. Let g be a polynomial that interlaces eachfi , and denote the largest root of g by α.

As fj has positive leading coefficient and any root of fj strictly tothe right of α has multiplicity 1, fj(α) ≤ 0 for all j . Thus

k∑j=1

fj(α) ≤ 0.

It follows that the largest real root β of∑k

j=1 fj satisfies β ≥ α.Hence fi (β) ≥ 0 for some i so that the largest root of fi lies in[α, β], because fi has exactly one root that is at least α. �





Definition

Let S1, . . . ,Sm be finite sets and let, for every (s1, . . . , sm) ∈S1 × · · · × Sm, fs1,...,sm(x) be a real-rooted polynomial of degree nwith positive leading coefficient. Given (s1, . . . , sk) ∈ S1 × · · · × Sk ,

fs1,...,sk :=∑

(sk+1,...,sm)∈Sk+1×···×Sm

fs1,...,sm .

f∅ :=∑

(s1,...,sm)∈S1×···×Sm

fs1,...,sm .

We say that the family {fs1,...,sm(x)}(s1,...,sm)∈S1×···×Sm is interlacingif, given 0 ≤ k ≤ m − 1 and (s1, . . . , sk) ∈ S1 × · · · × Sk , thepolynomials {fs1,...,sk ,t(x)}t∈Sk+1

have a common interlacing.





The following theorem follows immediately from the lemma.

Theorem

If {fs1,...,sm(x)}(s1,...,sm)∈S1×···×Sm is an interlacing family, then thereexists (s1, . . . , sm) ∈ S1 × · · · × Sm such that the largest root offs1,...,sm(x) is at most the largest (real) root of f∅.

In the proof that the family of characteristic polynomials fs(x) is aninterlacing family, we will use the following lemma (cf. Fisk ’08).

Lemma

Let f (x) and g(x) be (monic) real-rooted polynomials of degree nsuch that, for all λ ∈ [0, 1], the polynomial λf + (1− λ)g has nreal roots. Then f (x) and g(x) have a common interlacing.




Real-rooted polynomials: Towards Step 2 of the proof

We will next prove the following theorem, from which the mainresult will follow easily.

Theorem

Let p1, . . . , pm ∈ [0, 1] be given. Then the polynomial

∑s∈{±1}m

∏i : si=1

pi

∏i : si=−1

(1− pi )

fs(x)

is real-rooted.

Note: A polynomial over R is said to be real-rooted if it has nonon-real roots.




Real-rooted polynomials, cont.

We will use the notion of real stability:

Definition

Let f ∈ C[z1, . . . , zn] be given. We say that f is stable if f has nozeros in Hn, where H ⊂ C is the open upper half plane. We saythat f is real stable if f is stable and has real coefficients.

Remarks:

1◦ If f and g are (real) stable, then so is f ⊗ g .

2◦ By Hurwitz’ theorem from multivariate complex analysis, iff ∈ C[z1, . . . , zn] is (real) stable and c ∈ R, thenf (z1, . . . , zn−1, c) ∈ C[z1, . . . , zn−1] is (real) stable as well.

3◦ A univariate real polynomial is real stable if and only if it isreal-rooted.





We will need the following result:

Proposition (Lieb-Sokal ’81)

Assume that f (z1, . . . , zn) + yg(z1, . . . , zn) ∈ C[z1, . . . , zn, y ] isreal stable and has degree at most 1 in zj . Then f (z1, . . . , zn)−∂jg(z1, . . . , zn) is real stable.

Note: In order to prove this, it suffices to prove the statement with“real stable” replaced by “stable”. We follow Wagner ’11.

Lemma

Let f , g ∈ C[z1, . . . , zm] and assume that f is stable. Theng(z) + yf (z) is stable if and only if =(g(z)/f (z)) ≥ 0 for allz ∈ Hm.





Proof of the lemma. Assume that g(z) + yf (z) is stable and letz ∈ Hm be given. Then f (z) 6= 0 and we can find z0 ∈ C such thatg(z) + z0f (z) = 0. As g(z) + yf (z) is stable,

=(g(z)

f (z)

)= −=(z0) ≥ 0.

Conversely, if =(g(z)/f (z)) ≥ 0 for all z ∈ Hm andg(z) + z0f (z) = 0 for some z0 ∈ C and z ∈ Hm, then

=(z0) = −=(g(z)

f (z)

)≤ 0.

Thus g(z) + yf (z) has no zeros in Hm+1, i.e., is stable. �





Proof of the proposition. Assume that f (z) + yg(z) is stable andhas degree at most 1 in z1. Writing f (z) =

∑aI z

I , we get

yf (z1 − y−1, z2, . . . , zm)

=∑i1=0

aI yzI +

∑i1=1

aI (yzI − z i22 · · · z

imm ) = −∂1f (z) + yf (z).

As −y−1 ∈ H whenever y ∈ H, this polynomial is stable. By thelemma,

=(g(z)− ∂1f (z)

f (z)

)= =

(g(z)

f (z)

)+ =

(−∂1f (z)

f (z)

)≥ 0

for all z ∈ Hm. Hence g − ∂1f + yf is stable. The proof iscompleted by setting y = 0. �





Corollary

Assume f (z1, . . . , zn) and t(w1, . . . ,wm) are (real) stable, wherem ≤ n and both polynomials have degree at most 1 in thevariables zj ,wj for j = 1, . . . ,m. Then the polynomial

t (−∂1, . . . ,−∂m) f (z1, . . . , zn)

is (real) stable.

Proof. If m = 1 then t(w1) = α + βw1 so that αf (z) + yβf (z)= t(y)f (z) is stable. Thus t(−∂1)f (z) = αf (z)− β∂1f (z) isstable. If 2 ≤ m ≤ n, then t(−∂1, . . . ,−∂m−1, y)f (z) =t(−∂1, . . . ,−∂m−1, 0)f (z) + y(∂mt)(−∂1, . . . ,−∂m−1)f (z) isstable for every y ∈ H (by induction). Vary y to finish proof. �





Proposition (Borcea-Branden ’08)

Let A1, . . . ,Am be positive semidefinite matrices. Thendet[z1A1 + · · ·+ zmAm] is a real stable polynomial.

Proof. By Hurwitz’ theorem, we may assume that A1 is invertible.Moreover, it is clear that the polynomial has real coefficients.

Assume for contradiction that the polynomial has a zero(z1, . . . , zm) ∈ Hm. Write zj = βj + iλj , where λj > 0 and βj ∈ R.Put P = λ1A1 + · · ·+ λmAm and Q = β1A1 + · · ·+ βmAm. Then

0 = det(Q + iP) = det(P)det(P−1/2QP−1/2 + iI ),

i.e., −i ∈ σ(P−1/2QP−1/2), contradicting the fact that Q isself-adjoint. �





Theorem

Let a1, . . . , am, b1, . . . , bm ∈ Rn and p1, . . . , pm ∈ [0, 1] be given.Then the polynomial P(x) =

∑S⊆[m]

(∏i∈S

pi

)(∏i /∈S

(1− pi )

)det

[xI +

∑i∈S

aiaTi +

∑i /∈S

bibTi

]

is real-rooted.

Proof. Define a polynomial Q of 2m + 1 variables by

Q(x , u1, . . . , um, v1, . . . , vm) = det

[xI +

∑i∈S

uiaiaTi +

∑i /∈S

vibibTi

].

It follows from the previous proposition that Q is real stable.





Note that the degree of Q in each of the variables uj , vj is at most1. (It suffices to show this for the variable u1. As a1a

T1 is a rank 1

symmetric matrix, it follows that, as a function of u1 only, Q is ofthe form

det[u1a1aT1 + B] = det[u1E11(α) + B ′],

which is a polynomial of degree 1 by definition of det.)

Put Ti = ti (−∂ iu,−∂iv ), where ti (ui , vi ) = 1− piui − (1− pi )vi isreal stable. Then the corollary above implies that the operator

m∏i=1

Ti =∑

S ,T⊆[m]S∩T=∅

(∏i∈S

pi

)(∏i∈T

(1− pi )

)∂Su ∂

Tv

preserves the real stability of Q.Andreas Næs Aaserud An infinity of Ramanujan graphs




Finally, as substitution of real numbers preserves real stability, weget that the univariate polynomial

P(x) =

∑S ,T⊆[m]S∩T=∅

(∏i∈S

pi

)(∏i∈T

(1− pi )

)∂Su ∂

Tv Q(x , u, v)

|u1=···=um=0v1=···=vm=0

is real stable, hence real-rooted.

We claim that P = P. We will show this by comparing coefficients.

For this, we will need the formula

∂udet(uaaT + B) = det(aaT + B),

valid for any a ∈ Rm and B ∈Mm.





As the only terms that appear in P(x) are the ones that containexactly the variables with respect to which we are differentiating,we get that the coefficient of xd−k in P(x) is∑|R|+|W |=kR∩W=∅

(∏i∈R

pi

)(∏i∈W

(1− pi )

)det

[∑i∈R

aiaTi +

∑i∈W

bibTi

].

On the other hand, for S ⊆ [m], “Cauchy-Binet formula” implies

det

[xI +

∑i∈S

aiaTi +

∑i /∈S

bibTi

]

=d∑

k=0

xd−k∑|T |=k

det

∑i∈T∩S

aiaTi +

∑i∈T\S

bibTi

.Andreas Næs Aaserud An infinity of Ramanujan graphs




Thus

P(x) =d∑

k=0

xd−k∑

S⊆[m]

(∏i∈S

pi

)(∏i /∈S

(1− pi )

)

×∑|T |=k

det

∑i∈T∩S

aiaTi +

∑i∈T\S

bibTi

.Note also that, e.g. by induction on |[m] \ T |,∑

Q⊆[m]\T

∏i∈Q

pi∏

i∈([m]\T )\Q

(1− pi ) = 1.





Now we can write the coefficient of xd−k in P(x) as

∑|T |=k

∑R⊆T

det

∑i∈R

aiaTi +

∑i∈T\R

bibTi

×

∑Q⊆[m]\T

∏i∈Q∪R

pi

∏i∈([m]\Q)\R

(1− pi )

=∑|T |=k

∑R⊆T

det

∑i∈R

aiaTi +

∑i∈T\R

bibTi

(∏i∈R

pi

) ∏i∈T\R

(1− pi )

=

∑|R|+|W |=kR∩W=∅

(∏i∈R

pi

)(∏i∈W

(1− pi )

)det

[∑i∈R

aiaTi +

∑i∈W

bibTi

]. �





Theorem

Let p1, . . . , pm ∈ [0, 1] be given. Then the polynomial

∑s∈{±1}m

∏i : si=1

pi

∏i : si=−1

(1− pi )

fs(x)

is real-rooted.

Proof. Let d be the maximum degree of G . We will show that thepolynomial

Q(x) =∑

s∈{±1}m

∏i : si=1

pi

∏i : si=−1

(1− pi )

det(xI + dI − As)

has only real roots.Andreas Næs Aaserud An infinity of Ramanujan graphs




Denoting by eu the standard basis vector indexed by the vertex u,

dI−As =∑

(u,v)∈Esu,v=1

(eu−ev )(eu−ev )T +∑

(u,v)∈Esu,v=−1

(eu+ev )(eu+ev )T +D,

where D =∑

u∈V dueueTu is the diagonal matrix with entries

du = d − deg(u). Setting auv = eu − ev and buv = eu + ev ,

Q(x) =∑

s∈{±1}m

∏i : si=1

pi

∏i : si=−1

(1− pi )

× det

xI +∑u∈V

dueueTu +

∑(u,v)∈Esu,v=1

auvaTuv +

∑(u,v)∈Esu,v=−1

buvbTuv

.Andreas Næs Aaserud An infinity of Ramanujan graphs




Identify each s ∈ {±1}m with the set {i ∈ [m] : si = 1} and putpu = 1 for each u ∈ V . Then it is clear that the aforementionedpolynomial is of the same form as the one in the statement of theprevious theorem (restated below). Thus it is real-rooted. �

Theorem

Let a1, . . . , am, b1, . . . , bm ∈ Rn and p1, . . . , pm ∈ [0, 1] be given.Then the polynomial P(x) =

∑S⊆[m]

(∏i∈S

pi

)(∏i /∈S

(1− pi )

)det

[xI +

∑i∈S

aiaTi +

∑i /∈S

bibTi

]

is real-rooted.




Conclusion of Step 2: {fs(x)}s∈{±1}m is interlacing

Theorem

The family {fs(x)}s∈{±1}m is interlacing.

Proof. Let 0 ≤ k ≤ m − 1, (s1, . . . , sk) ∈ {±1}k , and λ ∈ [0, 1] begiven. We must show that the polynomial

λfs1,...,sk ,1(x) + (1− λ)fs1,...,sk ,−1(x)

is real-rooted. But this follows easily from the previous theoremwith p1 = (1 + s1)/2, . . . , pk = (1 + sk)/2, pk+1 = λ, andpk+2 = · · · = pm = 1/2. �




Steps 3 and 4: Wrapping up the proof

By applying the fact that {fs(x)}s∈{±1}m is interlacing, we getimmediately from the theorem of Heilmann-Lieb concerning theroots of the matching polynomial µG (x) that

Corollary

Let G be a d-regular bipartite graph with adjacency matrix A.Then G has a signing s such that every eigenvalue of As is at most2√d − 1. Thus, if G is a Ramanujan graph, so is the 2-lift

associated to s.

Corollary

Let d ≥ 3 be given. Then there is an infinite family of d-regularbipartite Ramanujan graphs.

Proof. Start with a complete graph and take successive 2-lifts. �




References

Main reference:

A. Marcus, Daniel A. Spielman, Nikhil Srivastava. InterlacingFamilies I: Bipartite Ramanujan graphs of all degrees. Preprint,2013 (available on arxiv).

Other references:

Yonatan Bilu and Nathan Linial. Lifts, discrepancy and nearlyoptimal spectral gap. Combinatorica 26 (5): pp. 495–519, 2006.

Julius Borcea and Petter Branden. Applications of stablepolynomials to mixed determinants: Johnson’s conjectures,unimodality, and symmetrized Fischer products. Duke Math. J. 143(2): pp. 205–223, 2008.

Steve Fisk. Polynomials, roots, and interlacing. Preprint, 2008(available on arxiv).

David G. Wagner. Multivariate stable polynomials: Theory andapplications. Bull. Amer. Math. Soc. 48 (1): pp. 53–84, 2011.


Documents

An infinity of Ramanujan graphs