Expanders and Ramanujan Graphs Mike Krebs, Cal State LA For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from A graph is a collection of vertices (dots) and edges (connections between vertices).

For slideshow: click “Research and Talks” from Distances don’t matter. All we care about is how the vertices are connected. These two graphs are the same, for example

For slideshow: click “Research and Talks” from Distances don’t matter. All we care about is how the vertices are connected. These two graphs are the same, for example

For slideshow: click “Research and Talks” from Consider the graph below.

For slideshow: click “Research and Talks” from The blue vertex below has degree 4.

For slideshow: click “Research and Talks” from The red vertex has degree 2.

For slideshow: click “Research and Talks” from So this graph is not “regular.”

For slideshow: click “Research and Talks” from Is this graph regular?

For slideshow: click “Research and Talks” from Yes, every vertex has degree 3.

For slideshow: click “Research and Talks” from We say it is 3-regular.

For slideshow: click “Research and Talks” from In this talk, we will be concerned primarily with regular graphs.

For slideshow: click “Research and Talks” from Think of a graph as a communications network. Two vertices can communcate directly with one another if they are connected by an edge.

For slideshow: click “Research and Talks” from Communication is instantaneous across edges, but there may be delays at vertices. Edges are expensive.

For slideshow: click “Research and Talks” from Goals: - Keep the degree fixed. - Let the number of vertices go to infinity. - Make sure the communications networks are “good.”

For slideshow: click “Research and Talks” from Main questions: How do we measure how good a graph is as a communications network? How good can we make them?

For slideshow: click “Research and Talks” from CAI H G F E D B J R U XZS TV W Y Q Here are two graphs. Each has 10 vertices. Each has degree 4.

For slideshow: click “Research and Talks” from Here are two graphs. Each has 10 vertices. Each has degree 4. Which one is a better communications network, and why? CAI H G F E D B J R U XZS TV W Y Q

For slideshow: click “Research and Talks” from I like the one on the right better. CAI H G F E D B J R U XZS TV W Y Q

For slideshow: click “Research and Talks” from You can get from any vertex to any other vertex in two steps. CAI H G F E D B J R U XZS TV W Y Q I like the one on the right better.

For slideshow: click “Research and Talks” from CAI H G F E D B J R U XZS TV W Y Q In the graph on the left, it takes at least three steps to get from A to F.

For slideshow: click “Research and Talks” from CAI H G F E D B J Let’s look at the set of vertices we can get to in n steps.

For slideshow: click “Research and Talks” from CAI H G F E D B J Let’s look at the set of vertices we can get to in n steps.

For slideshow: click “Research and Talks” from CAI H G F E D B J Here’s where we can get to in one step.

For slideshow: click “Research and Talks” from CAI H G F E D B J We would like to have many edges going outward from there.

For slideshow: click “Research and Talks” from CAI H G F E D B J Here’s where we can get to in two steps.

For slideshow: click “Research and Talks” from For any set S of vertices, we would like to have many edges from S to its complement.

For slideshow: click “Research and Talks” from CAI H G F E D B J The set of edges from S to its complement is called the boundary of S, denoted dS. Example: |S| = 5 |dS| = 6

For slideshow: click “Research and Talks” from Let G be a graph. Define h(G) to be the minimum value of over all sets S containing no more than half the vertices.

For slideshow: click “Research and Talks” from h(G) is called the expansion constant of G.

For slideshow: click “Research and Talks” from Example:

For slideshow: click “Research and Talks” from Example: = 2/1= 2

For slideshow: click “Research and Talks” from Example: = 2/2= 1

For slideshow: click “Research and Talks” from Example: = 4/2= 2

For slideshow: click “Research and Talks” from Example: 2/2=1 was the min. So h(X)=1.

For slideshow: click “Research and Talks” from The expansion constant tells us that even in a worst case, for any small set S, we have at least h(X)|S| edges going from S to its complement.

For slideshow: click “Research and Talks” from We want h(X) to be BIG! If a graph has small degree but many vertices, this is not easy.

For slideshow: click “Research and Talks” from Consider cycle graphs. They are 2-regular. Number of vertices goes to infinity.

For slideshow: click “Research and Talks” from Let’s see what happens to the expansion constants.

For slideshow: click “Research and Talks” from Let C be a cycle graph with n vertices.

For slideshow: click “Research and Talks” from Choose S to be the “bottom half.”

For slideshow: click “Research and Talks” from So S has n/2 or (n-1)/2 vertices.

For slideshow: click “Research and Talks” from But |dS|=2.

For slideshow: click “Research and Talks” from So h(C)<2/[(n-1)/2]=n-10

For slideshow: click “Research and Talks” from We say that a sequence of regular graphs is an expander family if:

For slideshow: click “Research and Talks” from We say that a sequence of regular graphs is an expander family if: (A) They all have the same degree.

For slideshow: click “Research and Talks” from We say that a sequence of regular graphs is an expander family if: (A) They all have the same degree. (2) The number of vertices goes to infinity.

For slideshow: click “Research and Talks” from We say that a sequence of regular graphs is an expander family if: (A) They all have the same degree. (2) The number of vertices goes to infinity. (iii) There exists a positive lower bound r such that the expansion constant is always at least r.

For slideshow: click “Research and Talks” from Expander families of degree 2 do not exist, as we just saw.

For slideshow: click “Research and Talks” from Expander families of degree 2 do not exist, as we just saw. Amazing fact: if d is any integer greater then 2, then an expander family of degree d exists.

For slideshow: click “Research and Talks” from Expander families of degree 2 do not exist, as we just saw. Amazing fact: if d is any integer greater then 2, then an expander family of degree d exists. (Constructing them is highly nontrivial!)

For slideshow: click “Research and Talks” from So far, we’ve looked at expansion from a combinatorial point of view. Now let’s look at it from an algebraic point of view.

We form the adjacency matrix of a graph as follows:

For slideshow: click “Research and Talks” from The expansion constant of a graph is closely related to the eigenvalues of its adjacency matrix.

For slideshow: click “Research and Talks” from Facts about graph eigenvalues:

For slideshow: click “Research and Talks” from Facts about graph eigenvalues: - the eigenvalues of a graph are all real

For slideshow: click “Research and Talks” from Facts about graph eigenvalues: - the eigenvalues of a graph are all real - the largest eigenvalue of a d-regular graph is d

For slideshow: click “Research and Talks” from Facts about graph eigenvalues: - the eigenvalues of a graph are all real - the largest eigenvalue of a d-regular graph is d - ifis the second largest eigenvalue, then

For slideshow: click “Research and Talks” from We call d-the spectral gap of the graph. h(G) is big iff the spectral gap is big.

For slideshow: click “Research and Talks” from A sequence of regular graphs is an expander family if: (A) They all have the same degree. (2) The number of vertices goes to infinity. (iii) There exists a positive lower bound r such that the spectral gap is always at least r.

For slideshow: click “Research and Talks” from We call d-the spectral gap of the graph. So we want the spectral gap to be big. So we wantto be small. But there is a limit to how small it can be.

For slideshow: click “Research and Talks” from must lie above the red curve.

For slideshow: click “Research and Talks” from at. The red curve had a horizontal asymptote

For slideshow: click “Research and Talks” from In other words, can be. at. is essentially the smallest The red curve had a horizontal asymptote

For slideshow: click “Research and Talks” from In other words, can be. at. is essentially the smallest We say a graph is Ramanujan if: < The red curve had a horizontal asymptote

For slideshow: click “Research and Talks” from Known: There exists a family of d-regular Ramanujan graphs if d-1 is a prime power.

For slideshow: click “Research and Talks” from Known: There exists a family of d-regular Ramanujan graphs if d-1 is a prime power. Open problem: Does a family of d-regular Ramanujan graphs for all d >3?

For slideshow: click “Research and Talks” from So... how do we construct these things?

For slideshow: click “Research and Talks” from One technique comes from group theory. Consider the integers mod 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} 6+8=4, for example

For slideshow: click “Research and Talks” from One technique comes from group theory. Consider the integers mod 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} This is a group.

For slideshow: click “Research and Talks” from One technique comes from group theory. Consider the integers mod 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} It has an identity element: 0+x=x for all x.

For slideshow: click “Research and Talks” from One technique comes from group theory. Consider the integers mod 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Every element has an inverse: The inverse of 4 is 6, since 4+6=0.

For slideshow: click “Research and Talks” from One technique comes from group theory. Consider the integers mod 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Also, the associative property holds.

For slideshow: click “Research and Talks” from One technique comes from group theory. Consider the integers mod 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Take a symmetric subset of the group. (That is, if x is in the subset, then so is the inverse of x.)

For slideshow: click “Research and Talks” from One technique comes from group theory. Consider the integers mod 10: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Take a symmetric subset of the group. (That is, if x is in the subset, then so is the inverse of x.) Example: {1,2,8,9}

For slideshow: click “Research and Talks” from We’ll use the group and the subset to form a graph. Here’s how.

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} Each group element becomes a vertex.

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} Each group element becomes a vertex

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} The vertex 4 is connected to

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} The vertex 4 is connected to 1+4=

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} The vertex 4 is connected to 2+4=

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} The vertex 4 is connected to 8+4=

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} The vertex 4 is connected to 9+4=

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} (That’s why we need a symmetric subset.)

For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} Continue this way to get...

0 For slideshow: click “Research and Talks” from Group: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Subset: {1,2,8,9} Continue this way to get

0 For slideshow: click “Research and Talks” from A graph constructed this way, from a group and a symmetric subset, is called a Cayley graph

0 For slideshow: click “Research and Talks” from The original Ramanujan families were constructed in this way

0 For slideshow: click “Research and Talks” from The original Ramanujan families were constructed in this way. However

0 For slideshow: click “Research and Talks” from The integers mod n gives us one example of a commutative group: a+b=b+a for all a,b

0 For slideshow: click “Research and Talks” from It is a theorem that Cayley graphs of commutative groups will never produce expander families

For slideshow: click “Research and Talks” from There are several other group-theoretic restrictions (solvable, bounded derived length, amenable).

For slideshow: click “Research and Talks” from There are several other group-theoretic restrictions (solvable, bounded derived length, amenable). It is known that all but two infinite families of finite simple groups can be made into expanders.

For slideshow: click “Research and Talks” from There are several other group-theoretic restrictions (solvable, bounded derived length, amenable). It is known that all but two infinite families of finite simple groups can be made into expanders. The last case to be done was the family of alternating groups. (Kassabov, 2007)