Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees

Adam Marcus Daniel Spielman Nikhil Srivastava

The expected characteristic polynomial of a random matrix

Interlacing Families of Polynomials

EM

[ �M (x) ]

The expected characteristic polynomial of a random matrix

is usually useless.

Interlacing Families of Polynomials

EM

[ �M (x) ]

The expected characteristic polynomial of a random matrix

EA[ pA(x) ]

is usually useless.

Interlacing Families of Polynomials

As are most expected polynomials

EM

[ �M (x) ]

Interlacing Families of Polynomials

But, if is an interlacing family, {pA(x)}A

max-root (pA(x)) max-root

⇣EA[ pA(x) ]

⌘

there exists an A so that

Expander Graphs

Regular graphs with many properties of random graphs.

Random walks mix quickly.

Every set of vertices has many neighbors.

Pseudo-random generators.

Error-correcting codes.

Hammer of Theoretical Computer Science.

Ramanujan Graphs

The spectrally best expanders. Let G be a graph and A be its adjacency matrix

a  

c  

d  

e  b  

         0          1          0          0          1            1          0          1          0          1            0          1          0          1          0            0          0          1          0          1            1          1          0          1          0  

Ramanujan Graphs

If G is d-regular, d is an eigenvalue of A If G is bipartite, the eigs of A are symmetric about 0; so -d is an eigenvalue too. d, -d are the trivial eigenvalues G is a good expander if all non-trivial eigenvalues are small

Ramanujan Graphs:

Alon-Boppana:

For all and for all sufficiently large n,

every d-regular graph with n vertices

has a non-trivial eigenvalue larger than

✏ > 0

2pd� 1� ✏

G is Ramanujan if all non-trivial eigenvalues have absolute value at most

2pd� 1

2pd� 1

Ramanujan Graphs Exist

complete bipartite graph: non-trivial eigenvalues = 0 complete graph: non-trivial eigenvalues = -1 The problem is to find infinite families. Margulis ‘88 and Lubotzky, Phillips, and Sarnak ’88 constructed infinite families of Ramanujan graphs. Friedman 08‘ proved random d-regular graphs are almost Ramanujan: 2

pd� 1 + ✏

Ramanujan Graphs Exist

All previously known constructions were regular of degree q+1, for a prime power q. We prove the existence of infinite families of bipartite Ramanujan graphs of every degree. And, are infinite families of (c,d)-biregular Ramanujan graphs, having non-trivial eigenvalues bounded by p

d� 1 +pc� 1

2-lifts of graphs

a  

c  

d  

e  b  

2-lifts of graphs

a  

c  

d  

e  b  

a  

c  

d  

e  b  

duplicate every vertex

2-lifts of graphs

duplicate every vertex

a0  

c0  

d0  

e0  

b0  

a1  

c1  

d1  

e1  

b1  

2-lifts of graphs

a0  

c0  

d0  

e0  

b0  

a1  

d1  

e1  

b1  

c1  

for every pair of edges: leave on either side (parallel), or make both cross

2-lifts of graphs

for every pair of edges: leave on either side (parallel), or make both cross

a0  

c0  

d0  

e0  

b0  

a1  

d1  

e1  

b1  

c1  

2-lifts of graphs

         0          1          0          0          1            1          0          1          0          1            0          1          0          1          0            0          0          1          0          1            1          1          0          1          0  

2-lifts of graphs

         0          1          0          0          1              0          0          0          0          0            1          0          1          0          1              0          0          0          0          0            0          1          0          1          0              0          0          0          0          0            0          0          1          0          1              0          0          0          0          0            1          1          0          1          0              0          0          0          0          0              0          0          0          0          0              0          1          0          0          1            0          0          0          0          0              1          0          1          0          1            0          0          0          0          0              0          1          0          1          0            0          0          0          0          0              0          0          1          0          1            0          0          0          0          0              1          1          0          1          0  

2-lifts of graphs

         0          0          0          0          1              0          1          0          0          0            0          0          1          0          1              1          0          0          0          0            0          1          0          0          0              0          0          0          1          0            0          0          0          0          1              0          0          1          0          0            1          1          0          1          0              0          0          0          0          0              0          1          0          0          0              0          0          0          0          1            1          0          0          0          0              0          0          1          0          1            0          0          0          1          0              0          1          0          0          0            0          0          1          0          0              0          0          0          0          1            0          0          0          0          0              1          1          0          1          0  

Eigenvalues of 2-lifts (Bilu-Linial)

Given a 2-lift of G, create a signed adjacency matrix As with a -1 for crossing edges and a 1 for parallel edges

         0        -­‐1          0          0          1          -­‐1          0          1          0          1            0          1          0        -­‐1          0            0          0        -­‐1          0          1            1          1          0          1          0  

a0  

c0  

d0  

e0  

b0  

a1  

d1  

e1  

b1  

c1  

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old)

and the eigenvalues of As (new) Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues

have absolute value at most 2pd� 1

Eigenvalues of 2-lifts (Bilu-Linial)

Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues

have absolute value at most 2pd� 1

Would give infinite families of Ramanujan Graphs: start with the complete graph, and keep lifting.

Eigenvalues of 2-lifts (Bilu-Linial)

Conjecture: Every d-regular graph has a 2-lift in which all the new eigenvalues

have absolute value at most 2pd� 1

We prove this in the bipartite case.

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: Every d-regular graph has a 2-lift in which all the new eigenvalues

have absolute value at most 2pd� 1

Trick: eigenvalues of bipartite graphs are symmetric about 0, so only need to bound largest

Eigenvalues of 2-lifts (Bilu-Linial)

Theorem: Every d-regular bipartite graph has a 2-lift in which all the new eigenvalues

have absolute value at most 2pd� 1

The expected polynomial

Specify a lift by

max-root (�As(x)) 2pd� 1

Prove is an interlacing family Conclude there is an s so that

�As(x)

Prove max-root

⇣Es[ �As(x) ]

⌘ 2

pd� 1

s 2 {±1}m

The expected polynomial

Es[ �As(x) ] = µG(x)

Theorem:

the matching polynomial of G

The matching polynomial (Heilmann-Lieb ‘72)

mi = the number of matchings with i edges

µG(x) =X

i�0

x

n�2i(�1)imi

µG(x) = x

6 � 7x4 + 11x2 � 2

µG(x) = x

6 � 7x4 + 11x2 � 2

one matching with 0 edges

µG(x) = x

6 � 7x4 + 11x2 � 2

7 matchings with 1 edge

µG(x) = x

6 � 7x4 + 11x2 � 2

µG(x) = x

6 � 7x4 + 11x2 � 2

The matching polynomial (Heilmann-Lieb ‘72)

µG(x) =X

i�0

x

n�2i(�1)imi

Theorem (Heilmann-Lieb) all the roots are real

The matching polynomial (Heilmann-Lieb ‘72)

µG(x) =X

i�0

x

n�2i(�1)imi

Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most

2pd� 1

Interlacing

Polynomial

interlaces

if

p(x) =Qn

i=1(x� ↵i)

q(x) =Qn�1

i=1 (x� �i)

↵1 �1 ↵2 · · ·↵n�1 �n�1 ↵n

Interlacing

Polynomial

interlaces

if

p(x) =Qn

i=1(x� ↵i)

q(x) =Qn�1

i=1 (x� �i)

↵1 �1 ↵2 · · ·↵n�1 �n�1 ↵n

and have a common interlacing

if there is a that interlaces both

p0(x) p1(x)

q(x)

Interlacing

If p0 and p1 have a common interlacing,

max-root (pi) max-root (Ei [ pi ])

for some i.

Interlacing

If p0 and p1 have a common interlacing,

max-root (pi) max-root (Ei [ pi ])

for some i.

largest  root  of  interlacer  

Without a common interlacing

Text

Without a common interlacing

Interlacing Family of Polynomials

is an interlacing family

if can be placed on the leaves of a tree so that when every node is the sum of leaves below, sets of siblings have common interlacings

p00

p01

p10

p11

p00 + p01

p10 + p11

{ps}s2{0,1}m

Interlacing Family of Polynomials

is an interlacing family

if can be placed on the leaves of a tree so that when every node is the sum of leaves below, sets of siblings have common interlacings

p00

p01

p10

p11

p0

p1

p;

{ps}s2{0,1}m

Interlacing Family of Polynomials p00

p01

p10

p11

p0

p1Theorem: There is an s so that

Ps2{0,1}m ps = p;

max-root (ps) max-root

�Es2{0,1}m [ ps ]

�

An interlacing family Theorem: Let ps(x) = �As(x)

is an interlacing family

Lemma (old, see Fisk): and have a common interlacing

if and only if

is real rooted for all

p0(x) p1(x)

�p0(x) + (1� �)p1(x)

0 � 1

{ps}s2{±1}m

An interlacing family Lemma (old, see Fisk): and have a common interlacing

if and only if

is real rooted for all

p0(x) p1(x)

�p0(x) + (1� �)p1(x)

0 � 1

Establish Real Rootedness through theory of Real Stable Polynomials

Godsil’s Proof of Heilmann-Lieb

T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step

Godsil’s Proof of Heilmann-Lieb

a  

c  

d  

e  b  

a  

ab

e

a  

e

ab

e

ab

c

ab

d

e

a

Godsil’s Proof of Heilmann-Lieb

T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step

Theorem: The matching polynomial divides the characteristic polynomial of T(G,v)

Godsil’s Proof of Heilmann-Lieb

For (c,d)-regular, implies pd� 1 +

pc� 1

T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step

Theorem: The matching polynomial divides the characteristic polynomial of T(G,v)

Question

What else can you characterize, and show exists, using polynomials?