Download presentation
Presentation is loading. Please wait.
1
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
2
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Expanders and Ramanujan Graphs Mike Krebs Cal State LA For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
3
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of a graph For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
4
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of a graph For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
5
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Think of a graph as a communications network. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
6
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Two vertices can communcate directly with one another
7
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Two vertices can communcate directly with one another if they are connected by an edge. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
8
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Communication is instantaneous across edges, but there may be delays at vertices. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
9
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Edges are expensive. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
10
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs In this talk, we will be concerned primarily with regular graphs. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
11
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs That is, same degree (number of edges) at each vertex. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
12
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Goals:
13
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Goals: ● Keep the degree fixed
14
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Goals: ● Let the number of vertices go to infinity. ● Keep the degree fixed
15
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs ● Make sure the communications networks are as good as possible. For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs ● Let the number of vertices go to infinity. Goals: ● Keep the degree fixed
16
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Main questions: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
17
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Main questions: How do we measure how good a graph is as a communications network?
18
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs How good can we make them? For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs How do we measure how good a graph is as a communications network? Main questions:
19
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
20
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
21
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
22
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
23
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
24
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
25
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s where we can get to in one step.
26
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Here’s where we can get to in one step.
27
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
28
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
29
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
30
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
31
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs CAI H G F E D B J For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
32
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
33
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
34
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
35
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
36
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
37
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We want h(X) to be BIG! For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
38
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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 www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
39
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider cycle graphs.
40
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider cycle graphs. They are 2-regular.
41
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Consider cycle graphs. They are 2-regular. Number of vertices goes to infinity.
42
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let’s see what happens to the expansion constants.
43
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Let S be the “bottom half”...
44
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
45
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We say that a sequence of regular graphs is an expander family if:
46
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We say that a sequence of regular graphs is an expander family if: (A) They all have the same degree.
47
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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.
48
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (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.
49
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Expander families of degree 2 do not exist, as we just saw.
50
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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.
51
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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
52
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs 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.
53
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs We form the adjacency matrix of a graph as follows:
54
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs The expansion constant of a graph is closely related to the eigenvalues of its adjacency matrix.
55
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about eigenvalues of a d-regular graph G:
56
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about eigenvalues of a d-regular graph G: ● They are all real.
57
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Facts about eigenvalues of a d-regular graph G: ● They are all real. ● The largest eigenvalue is d.
58
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs ● If Facts about eigenvalues of a d-regular graph G: For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs is the second largest eigenvalue, then (Alon-Dodziuk-Milman-Tanner) ● They are all real. ● The largest eigenvalue is d.
59
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (Alon-Dodziuk-Milman-Tanner)
60
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (Alon-Dodziuk-Milman-Tanner)
61
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs (Alon-Dodziuk-Milman-Tanner)
62
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network. Take-home Message #2:
63
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
64
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
65
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs.
66
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs.
67
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs.
68
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs.
69
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs Take-home Message #1: The expansion constant is one measure of how good a graph is as a communications network.
70
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
71
For slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebsFor slideshow: click “Research and Talks” from www.calstatela.edu/faculty/mkrebswww.calstatela.edu/faculty/mkrebs
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.