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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
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 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 matching polynomial (Heilmann-Lieb ‘72)
mi = the number of matchings with i edges
µG(x) =X
i�0
x
n�2i(�1)imi
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.
largest root of interlacer
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
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)