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

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs


Expanders and Ramanujan Graphs

Mike Krebs

Cal State LA

For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs


Think of a graph



Think of a graph



Think of a graph as acommunications network.


For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

Two vertices can communcatedirectly with one another


Two vertices can communcatedirectly with one another ifthey are connected by an edge.



Communication is instantaneousacross edges, but there may bedelays at vertices.



Edges are expensive.



In this talk, we will be concernedprimarily with regular graphs.



That is, same degree (numberof edges) at each vertex.



Goals:


Goals:

● Keep the degree fixed


Goals:

● Let the number of vertices go to infinity.



● Make sure the communications networks are as good as possible.For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebs

● Let the number of vertices go to infinity.

Goals:



Main questions:



Main questions:

How do we measure how gooda graph is as a communicationsnetwork?


How good can we make them?


How do we measure how gooda graph is as a communicationsnetwork?

Main questions:


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Here are two graphs. Each has 10 vertices. Each has degree 4.



Here are two graphs. Each has 10 vertices. Each has degree 4.

Which one is a better communications network, and why?

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I like the one on the right better.

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You can get from any vertex to any other vertex in two steps.

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I like the one on the right better.



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In the graph on the left, it takes at least three steps to get fromA to F.



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Let’s look at the set of vertices we can get to in n steps.



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Here’s where we can get to in one step.


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Here’s where we can get to in one step.


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We would like to have many edges going outward from there.



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Here’s where we can get to in two steps.




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Take-home Message #1:

The expansion constantis one measure of howgood a graph is as acommunications network.



We want h(X) to be BIG!



We want h(X) to be BIG!

If a graph has small degreebut many vertices, this is noteasy.



Consider cycle graphs.


Consider cycle graphs.They are 2-regular.


Consider cycle graphs.They are 2-regular.Number of vertices goes toinfinity.


Let’s see what happens tothe expansion constants.


Let S be the “bottom half” . . .



We say that a sequence ofregular graphs is an expanderfamily 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.



(2) The number of vertices goes to infinity.


Expander families of degree 2 do not exist,as we just saw.



Amazing fact: if d is any integer greaterthen 2, then an expander family of degreed exists.



Amazing fact: if d is any integer greaterthen 2, then an expander family of degreed exists. (Constructing them explicitlyis highly nontrivial!)

Existence: Pinsker 1973First explicit construction: Margulis 1973


So far, we’ve looked at expansion froma combinatorial point of view.

Now let’s look at it from an algebraic pointof view.


We form theadjacency matrixof a graph as follows:


The expansion constant of a graph isclosely related to the eigenvalues ofits adjacency matrix.


Facts about eigenvalues of a d-regulargraph G:



● They are all real.




● The largest eigenvalue is d.


● If



is the second largest eigenvalue, then

(Alon-Dodziuk-Milman-Tanner)


● The largest eigenvalue is d.








Take-home Message #1: The expansion constant is onemeasure of how good a graph is as a communications network.

Take-home Message #2:




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Take-home Message #1: The expansion constant is onemeasure of how good a graph is as a communications network.



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

Documents

communications network