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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Expanders and Ramanujan Graphs For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Think of a graph For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Think of a graph For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Think of a graph as a communications network. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Two vertices can communicate directly with one another

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Two vertices can communicate directly with one another iff they are connected by an edge. For slideshow: click “Research and Talks” from

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Edges are expensive. For slideshow: click “Research and Talks” from

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from That is, same degree (number of edges) at each vertex. For slideshow: click “Research and Talks” from

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Goals: ● Keep the degree fixed

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Goals: ● Let the number of vertices go to infinity. ● Keep the degree fixed

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Main questions: For slideshow: click “Research and Talks” from

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

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Remark: For regular graphs, this “communications network” business is more of an analogy than an application. But expanders have many real-world applications, including: Cryptographic hash functions Structural engineering Error-correcting codes Derandomization and many more...

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Which one is a better communications network, and why? For slideshow: click “Research and Talks” from Consider the two graphs below. Each has 46 vertices and is 3-regular.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from In some sense, the graph on the left is better; you can get from any vertex to any other vertex in 8 steps or fewer. (The diameter is 8.) For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Whereas in the graph on the right, you need at least 11 steps to get from point 1 to point 12.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from So the graph on the left is a faster network. (It has a smaller diameter.) For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from The graph on the left has a fatal flaw, though. Deleting just one edge separates 15 of the vertices from the other 31 vertices. (It has connectivity 1.) For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from You can delete any two edges from the graph on the right, though, and it will still be connected. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from So the graph on the right is a more reliable network. (It has larger connectivity.) For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Remarkably, we can capture both of these concepts with a single measurement. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Let’s look at the set of vertices we can get to in n steps. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Here’s where we can get to in one step.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Here’s where we can get to in one step.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Two steps.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Two steps.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from At each stage, we would like to have many edges going outward from the points we’ve been to so far. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from But we run into “overlap”---many red vertices but relatively few green edges. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Similarly for the other graph. For slideshow: click “Research and Talks” from

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from CAI H G F E D B J For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from CAI H G F E D B J For slideshow: click “Research and Talks” from

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

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

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

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from The expansion constant captures both the speed and the reliability of the communications network. For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from We want h(X) to be BIG! For slideshow: click “Research and Talks” from

For slideshow: click “Research and Talks” from 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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Consider cycle graphs.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Consider cycle graphs. They are 2-regular.

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

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

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from 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 slideshow: click “Research and Talks” from 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 slideshow: click “Research and Talks” from 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 slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from (iii) There exists a positive lower bound r such that the expansion constant is always at least r. 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 slideshow: click “Research and Talks” from 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 slideshow: click “Research and Talks” from 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 slideshow: click “Research and Talks” from 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 explicitly is highly nontrivial!) Existence: Pinsker 1973 First explicit construction: Margulis 1973

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

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from COMING THIS FALL!!!

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Shameless self-promotion!!! COMING THIS FALL!!!

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from 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 another point of view.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from We form the adjacency matrix of a graph as follows:

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from 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 slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Facts about eigenvalues of a d-regular graph G:

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Facts about eigenvalues of a d-regular graph G: ● They are all real.

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from ● If Facts about eigenvalues of a d-regular graph G: For slideshow: click “Research and Talks” from is the second largest eigenvalue, then (Alon-Dodziuk-Milman-Tanner) ● They are all real. ● The largest eigenvalue is d.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from (Alon-Dodziuk-Milman-Tanner)

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from (Alon-Dodziuk-Milman-Tanner)

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from (Alon-Dodziuk-Milman-Tanner)

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network. Take-home Message #2:

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Student Research Projects

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Student Research Projects One way to learn about eigenvalues: count closed walks in the graph.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Student Research Projects One way to learn about eigenvalues: count closed walks in the graph. Number of closed walks of length n equals the sum of the nth powers of the eigenvalues.

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Student Research Projects Matthew AivazianKeith Dsouza Natalie Martinez Certain Cayley graphs Cycle graphs Small graphs

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Shameless self-promotion!!! COMING THIS FALL!!!

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Take-home Message #2:

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

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

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

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

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

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

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from (Actually, this definition is slightly inaccurate, but never mind...)

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

For slideshow: click “Research and Talks” from slideshow: click “Research and Talks” from Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network.

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

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

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