Ramanujan Graphs of Every Degree Adam Marcus (Crisply, Yale)
Daniel Spielman (Yale) Nikhil Srivastava (MSR India)
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Expander Graphs Sparse, regular well-connected graphs with many
properties of random graphs. Random walks mix quickly. Every small
set of vertices has many neighbors. Pseudo-random generators.
Error-correcting codes. Sparse approximations of complete graphs.
Major theorems in Theoretical Computer Science.
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Spectral 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
Eigenvalues If d -regular (every vertex has d edges), trivial
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Spectral Expanders If bipartite (all edges between two
parts/colors) eigenvalues are symmetric about 0 trivial a cd f b e
If d -regular and bipartite, 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1
1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0
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Spectral Expanders 0 [] -d d G is a good spectral expander if
all non-trivial eigenvalues are small
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Bipartite Complete Graph Adjacency matrix has rank 2, so all
non-trivial eigenvalues are 0 a cd f b e 0 0 0 1 1 1 1 1 1 0 0
0
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Spectral Expanders 0 [] -d d Challenge: construct infinite
families of fixed degree G is a good spectral expander if all
non-trivial eigenvalues are small
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Spectral Expanders G is a good spectral expander if all
non-trivial eigenvalues are small 0 [] -d d Challenge: construct
infinite families of fixed degree Alon-Boppana 86: Cannot beat )
(
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G is a Ramanujan Graph if absolute value of non-trivial eigs 0
[] -d d ) ( Ramanujan Graphs:
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G is a Ramanujan Graph if absolute value of non-trivial eigs 0
[] -d d ) ( Ramanujan Graphs: Margulis, Lubotzky-Phillips-Sarnak88:
Infinite sequences of Ramanujan graphs exist for
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G is a Ramanujan Graph if absolute value of non-trivial eigs 0
[] -d d ) ( Ramanujan Graphs: Margulis, Lubotzky-Phillips-Sarnak88:
Infinite sequences of Ramanujan graphs exist for Can be quickly
constructed: can compute neighbors of a vertex from its name
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G is a Ramanujan Graph if absolute value of non-trivial eigs 0
[] -d d ) ( Ramanujan Graphs: Friedman08: A random d -regular graph
is almost Ramanujan :
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Ramanujan Graphs of Every Degree Theorem: there are infinite
families of bipartite Ramanujan graphs of every degree.
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Theorem: there are infinite families of bipartite Ramanujan
graphs of every degree. And, are infinite families of (c,d)
-biregular Ramanujan graphs, having non-trivial eigs bounded by
Ramanujan Graphs of Every Degree
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Find an operation that doubles the size of a graph without
creating large eigenvalues. 0 [] -d d Bilu-Linial 06 Approach (
)
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0 [] -d d Bilu-Linial 06 Approach Find an operation that
doubles the size of a graph without creating large eigenvalues. (
)
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2-lifts of graphs a c d e b
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a c d e b a c d e b duplicate every vertex
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2-lifts of graphs duplicate every vertex a0a0 c0c0 d0d0 e0e0
b0b0 a1a1 c1c1 d1d1 e1e1 b1b1
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2-lifts of graphs a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1
c1c1 for every pair of edges: leave on either side (parallel), or
make both cross
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2-lifts of graphs for every pair of edges: leave on either side
(parallel), or make both cross a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1
e1e1 b1b1 c1c1
Eigenvalues of 2-lifts (Bilu-Linial) Given a 2-lift of G,
create a signed adjacency matrix A s 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 a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1
c1c1
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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 A s (new) 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 a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1
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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 Theorem: The eigenvalues of the 2-lift are
the union of the eigenvalues of A (old) and the eigenvalues of A s
(new)
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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 Would give infinite families of Ramanujan
Graphs: start with the complete graph, and keep lifting.
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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 We prove this in the bipartite case. a
2-lift of a bipartite graph is bipartite
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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 Trick: eigenvalues of bipartite graphs are symmetric
about 0, so only need to bound largest
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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
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First idea: a random 2-lift Specify a lift by Pick s uniformly
at random
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First idea: a random 2-lift Specify a lift by Pick s uniformly
at random Are graphs for which this usually fails
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First idea: a random 2-lift Specify a lift by Pick s uniformly
at random Are graphs for which this usually fails Bilu and Linial
proved G almost Ramanujan, implies new eigenvalues usually small.
Improved by Puder and Agarwal-Kolla-Madan
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The expected polynomial Consider
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The expected polynomial Consider Conclude there is an s so that
Prove Prove is an interlacing family
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The expected polynomial Theorem (Godsil-Gutman 81): the
matching polynomial of G
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The matching polynomial (Heilmann-Lieb 72) m i = the number of
matchings with i edges
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one matching with 0 edges
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7 matchings with 1 edge
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations same edge: same
value
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations same edge: same
value
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations Get 0 if hit any
0s
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations Get 0 if take just
one entry for any edge
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations Only permutations
that count are involutions
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations Only permutations
that count are involutions
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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0
0 1 x 1 1 0 0 0 1 x Expand using permutations Only permutations
that count are involutions Correspond to matchings
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The matching polynomial (Heilmann-Lieb 72) Theorem
(Heilmann-Lieb) all the roots are real
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The matching polynomial (Heilmann-Lieb 72) Theorem
(Heilmann-Lieb) all the roots are real and have absolute value at
most
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The matching polynomial (Heilmann-Lieb 72) Theorem
(Heilmann-Lieb) all the roots are real and have absolute value at
most Implies
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Interlacing Polynomial interlaces if
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Common Interlacing and have a common interlacing if can
partition the line into intervals so that each interval contains
one root from each poly
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Common Interlacing and have a common interlacing if can
partition the line into intervals so that each interval contains
one root from each poly ) ) ) ) ( (( (
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If p 1 and p 2 have a common interlacing, for some i. Largest
root of average Common Interlacing
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If p 1 and p 2 have a common interlacing, for some i. Largest
root of average Common Interlacing
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is an interlacing family If the polynomials can be placed on
the leaves of a tree so that when put average of descendants at
nodes siblings have common interlacings Interlacing Family of
Polynomials
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is an interlacing family Interlacing Family of Polynomials If
the polynomials can be placed on the leaves of a tree so that when
put average of descendants at nodes siblings have common
interlacings
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Interlacing Family of Polynomials There is an s so that
Theorem:
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An interlacing family Theorem: Let is an interlacing
family
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Interlacing and have a common interlacing iff is real rooted
for all
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To prove interlacing family Let
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To prove interlacing family Let Need to prove that for all, is
real rooted
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To prove interlacing family Let are fixed is 1 with
probability, -1 with are uniformly Need to prove that for all, is
real rooted
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Generalization of Heilmann-Lieb We prove is real rooted for
every independent distribution on the entries of s
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Generalization of Heilmann-Lieb We prove is real rooted for
every independent distribution on the entries of s By using mixed
characteristic polynomials
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Mixed Characteristic Polynomials For independently chosen
random vectors is their mixed characteristic polynomial. Theorem:
Mixed characteristic polynomials are real rooted. Proof: Using
theory of real stable polynomials.
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Mixed Characteristic Polynomials For independently chosen
random vectors is their mixed characteristic polynomial. Obstacle:
our matrix is a sum of random rank-2 matrices 0 -1 -1 0 0 1 1 0
or
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Mixed Characteristic Polynomials For independently chosen
random vectors is their mixed characteristic polynomial. Solution:
add to the diagonal 1 -1 -1 1 1 1 or
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Generalization of Heilmann-Lieb We prove is real rooted for
every independent distribution on the entries of s Implies is an
interlacing family
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Generalization of Heilmann-Lieb We prove is real rooted for
every independent distribution on the entries of s Conclude there
is an s so that Implies is an interlacing family
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Universal Covers The universal cover of a graph G is a tree T
of which G is a quotient. vertices map to vertices edges map to
edges homomorphism on neighborhoods Is the tree of non-backtracking
walks in G. The universal cover of a d -regular graph is the
infinite d -regular tree.
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Quotients of Trees d -regular Ramanujan as quotient of infinite
d -ary tree Spectral radius and norm of inf d -ary tree are
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Godsils 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
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Godsils Proof of Heilmann-Lieb a c d e b a a b e a e a b e a b
c a b d e a
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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)
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Godsils 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) Is a subgraph of infinite tree,
so has smaller spectral radius
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Quotients of Trees (c,d) -biregular bipartite Ramanujan as
quotient of infinite (c,d) -ary tree Spectral radius For (c,d)
-regular bipartite Ramanujan graphs
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Irregular Ramanujan Graphs (Greenberg-Lubotzky) Def: G is
Ramanujan if its non-trivial eigenvalues have abs value less than
the spectral radius of its cover Theorem: If G is bipartite and
Ramanujan, then there is an infinite family of Ramanujan graphs
with the same cover.
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Questions Non-bipartite Ramanujan Graphs of every degree?
Efficient constructions? Explicit constructions?
