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?