Cayley Graphs & Expanders Steve the presenter. Some history Arthur Cayley.

Slides:



Advertisements
Similar presentations
NP-Hard Nattee Niparnan.
Advertisements

Lecture 5 Graph Theory. Graphs Graphs are the most useful model with computer science such as logical design, formal languages, communication network,
Graph-02.
1 Appendix B: Solving TSP by Dynamic Programming Course: Algorithm Design and Analysis.
Algebraic Structures DEFINITIONS: PROPERTIES OF BINARY OPERATIONS Let S be a set and let  denote a binary operation on S. (Here  does not necessarily.
Symmetric Group Sym(n) As we know a permutation  is a bijective mapping of a set A onto itself:  : A  A. Permutations may be multiplied and form the.
Entropy Rates of a Stochastic Process
C++ Programming: Program Design Including Data Structures, Third Edition Chapter 21: Graphs.
DERIVATIVES ARE FUNCTIONS TOO ! First of all, let us see how many consequences are implied by the statement The derivative of at exists. Recall that the.
Section 2.1 Euler Cycles Vocabulary CYCLE – a sequence of consecutively linked edges (x 1,x2),(x2,x3),…,(x n-1,x n ) whose starting vertex is the ending.
International Workshop on Computer Vision - Institute for Studies in Theoretical Physics and Mathematics, April , Tehran 1 IV COMPUTING SIZE.
EXPANDER GRAPHS Properties & Applications. Things to cover ! Definitions Properties Combinatorial, Spectral properties Constructions “Explicit” constructions.
Math Foundations Week 12 Graphs (2). Agenda Paths Connectivity Euler paths Hamilton paths 2.
Thompson’s Group Jim Belk. Associative Laws Let  be the following piecewise-linear homeomorphism of  :
CS5371 Theory of Computation Lecture 1: Mathematics Review I (Basic Terminology)
Expanders Eliyahu Kiperwasser. What is it? Expanders are graphs with no small cuts. The later gives several unique traits to such graph, such as: – High.
Graphs Rosen 8.1, 8.2. There Are Many Uses for Graphs! Networks Data organizations Scene graphs Geometric simplification Program structure and processes.
GRAPH Learning Outcomes Students should be able to:
Theory of Computing Lecture 10 MAS 714 Hartmut Klauck.
 Jim has six children.  Chris fights with Bob,Faye, and Eve all the time; Eve fights (besides with Chris) with Al and Di all the time; and Al and Bob.
Graphs Chapter 10.
7 Graph 7.1 Even and Odd Degrees.
Nattee Niparnan. Easy & Hard Problem What is “difficulty” of problem? Difficult for computer scientist to derive algorithm for the problem? Difficult.
Discrete Mathematics, 1st Edition Kevin Ferland
Graph Theoretic Concepts. What is a graph? A set of vertices (or nodes) linked by edges Mathematically, we often write G = (V,E)  V: set of vertices,
An introduction to expander families and Ramanujan graphs
Cryptographic hash functions from expander graphs Denis Charles, Microsoft Research Eyal Goren, McGill University Kristin Lauter, Microsoft Research ECC.
4.1 Connectivity and Paths: Cuts and Connectivity
Discrete Math for CS Binary Relation: A binary relation between sets A and B is a subset of the Cartesian Product A x B. If A = B we say that the relation.
5.2 Trees  A tree is a connected graph without any cycles.
GRAPHS THEROY. 2 –Graphs Graph basics and definitions Vertices/nodes, edges, adjacency, incidence Degree, in-degree, out-degree Subgraphs, unions, isomorphism.
Graphs.  Definition A simple graph G= (V, E) consists of vertices, V, a nonempty set of vertices, and E, a set of unordered pairs of distinct elements.
Markov Chains and Random Walks. Def: A stochastic process X={X(t),t ∈ T} is a collection of random variables. If T is a countable set, say T={0,1,2, …
CS 3240 – Chapter 4.  Closure Properties  Algorithms for Elementary Questions:  Is a given word, w, in L?  Is L empty, finite or infinite?  Are L.
COSC 2007 Data Structures II Chapter 14 Graphs I.
Chapter 5 Graphs  the puzzle of the seven bridge in the Königsberg,  on the Pregel.
Math 3121 Abstract Algebra I Lecture 5 Finish Sections 6 + Review: Cyclic Groups, Review.
An Introduction to Graph Theory
Graph Theory and Applications
Lecture 10: Graph-Path-Circuit
Math 3121 Abstract Algebra I Lecture 7: Finish Section 7 Sections 8.
CS 203: Introduction to Formal Languages and Automata
Recognising Languages We will tackle the problem of defining languages by considering how we could recognise them. Problem: Is there a method of recognising.
Trees Thm 2.1. (Cayley 1889) There are nn-2 different labeled trees
8.4 Closures of Relations Definition: The closure of a relation R with respect to property P is the relation obtained by adding the minimum number of.
SECTION 8 Groups of Permutations Definition A permutation of a set A is a function  ϕ : A  A that is both one to one and onto. If  and  are both permutations.
Introduction to Graph Theory
Chapter 9: Graphs.
Introduction to Graph Theory
CS 261 – Nov. 17 Graph properties – Bipartiteness – Isomorphic to another graph – Pseudograph, multigraph, subgraph Path Cycle – Hamiltonian – Euler.
Chromatic Coloring with a Maximum Color Class Bor-Liang Chen Kuo-Ching Huang Chih-Hung Yen* 30 July, 2009.
12. Lecture WS 2012/13Bioinformatics III1 V12 Menger’s theorem Borrowing terminology from operations research consider certain primal-dual pairs of optimization.
Introduction to NP Instructor: Neelima Gupta 1.
Grade 11 AP Mathematics Graph Theory Definition: A graph, G, is a set of vertices v(G) = {v 1, v 2, v 3, …, v n } and edges e(G) = {v i v j where 1 ≤ i,
1 GRAPH Learning Outcomes Students should be able to: Explain basic terminology of a graph Identify Euler and Hamiltonian cycle Represent graphs using.
Theory of Computational Complexity Probability and Computing Ryosuke Sasanuma Iwama and Ito lab M1.
Lecture 20. Graphs and network models 1. Recap Binary search tree is a special binary tree which is designed to make the search of elements or keys in.
Expanders and Ramanujan Graphs Mike Krebs, Cal State LA For slideshow: click “Research and Talks” from
Math 3121 Abstract Algebra I Lecture 6 Midterm back over+Section 7.
Chapter Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity.
An Introduction to Graph Theory
Applied Discrete Mathematics Week 13: Graphs
Special Graphs By: Sandeep Tuli Astt. Prof. CSE.
CSE 373, Copyright S. Tanimoto, 2001 Graphs 1 -
Grade 11 AP Mathematics Graph Theory
Taibah University College of Computer Science & Engineering Course Title: Discrete Mathematics Code: CS 103 Chapter 10 Graphs Slides are adopted from “Discrete.
CSE 373, Copyright S. Tanimoto, 2002 Graphs 1 -
Applied Combinatorics, 4th Ed. Alan Tucker
HW 3 (Due Wednesday Feb 6) Create slide(s) for your 1 minute presentation on a graph theory application. Make sure your slide(s) include (1) Define the.
HW 3 (Due Wednesday Feb 6) Create slide(s) for your 1 minute presentation on a graph theory application. Make sure your slide(s) include (1) Define the.
Presentation transcript:

Cayley Graphs & Expanders Steve the presenter

Some history Arthur Cayley

Son of a British merchant in Russia who wedded a Russian girl. English Super Genius Did everything Very smart and knew more than 6 languages including yours and math. Occupied Lucasian Chair and fathered British pure math Died at 74 The Real Arthur Cayley For more demoralizing facts, visit his homepage on Wikipedia

Some Definitions Let G be a graph (finite and infinite), and let S be a non-empty, finite subset of G. Here we assume S to be symmetric, S = S −1 Definition: The Cayley graph G (G, S) is the graph with the vertex set V = G and edge set E = {{x,y} : x,y in G, there exist s in S : y = xs}

Hence, two vertices are adjacent if one is obtained from the other by right multiplication by some element of S. Note, since S is symmetric, this adjacency relation is also symmetric, i.e, a ~ b implies b ~ a, so that the resulting graphs are undirected.

Cayley graphs are motivated by the Cayley Theorem, which states hat every group G is isomorphic to a subgroup of the symmetric group on G.

Let's look at some examples: If G = Zn is the finite cyclic group of order n and the set S consists of two elements, the standard generator of G and its inverse, then the Cayley graph is the cycle Cn.

G = Z/6Z, S = {1, -1}

G = Z/6Z, S = {2, -2}

G = Z/6Z, S = {3}

G = Z/6Z, S = {2, -2, 3}

G = Sym(3), S = {(123), (132), (12)}

non-isomorphic groups, isomorphic Cayley Graphs

MOre Examples Similarly, suppose now that G = Z is the infinite cyclic group and the set S consists of the standard generator 1 and its inverse (−1 in the additive notation) then the Cayley graph is an infinite chain.

G = Z, S = {1, -1}

This of course simply the number line.

G = Z, S = {2, -2, 3, -3}

G = Z 2, S = {(0,1), (-1,0), (0,1), (0,-1)}

G = L2, the free group on 2 generators a, b; S = {a, a-1, b, b-1}

Alternatively, if we draw each new edge horizontal and vertical to half the length, this gives rise to fractal images. The Cayley graphs of infinite groups provide interesting geometries!

Expanders Now what is an expander?

The investigation of an expander centres around the following question: Given a sparse set of points, how do we systematically construct a highly connected yet efficient network between the points? Conceivably, the answer to such a question is of great interest to electric grid constructors and designers of network between computers among many others.

Cayley Graphs, Groups and expanders In the previous slides, we have seen that Cayley graphs give very nice encoding of the structure of discrete groups. Conversely, it also helps us sometimes to translate a graph problem back into its group presentation. This way, we can I will not tell you how now because I don't know yet. But do remember to ask me about it if you see me next semester!

Expanders We consider graphs X = (V, E), where V again is the set of vertices and E the set of edges of X. Here we shall limit ourselves to the case that X is undirected and finite.

A path in X is a sequence v1, v2,...., vk of vertices, where vi is adjacent to vi+1 (i.e. and {vi, vi+1} is an edge). A graph X is connected if every two vertices can be joined by a path. (One can trace from one vertice to another one through a series of edges.)‏ Finally a graph is k-regular if every vertex is also connected to k other vertices through k-edges.

This is the celebrated Petersen Graph. It is clearly connected. In fact, it is a 3-regular expander.

Let F be a subset of V, The boundary ∂F is the the set of edges connecting F to V – F. In our case here, F = 3, V – F = 7, ∂F are the seven black “fat” edges.

The expanding constant, or the isoperimetric constant of X, is: h(X) = inf {abs(∂F)/(min{abs(F),abs(V – F)}) : F is subset of V, 0 < abs(F) < +∞}

If we view X as a network transmitting information (where retained by some vertex propagates, say in one unit of time, to neighboring vertices), then h(X) measures the quality of X as a network: h(X) is large, information propagates well.

Let's consider two extreme examples: The complete graph Km on m vertices is defined s.t. Every vertex is connected to one another, e.g. m = 5. It's clear in this case, if F = i, ∂F = i(m-i) s.t. h(Km) = m/2

The cycle Cn on n vertices: When n = 6, if F is a half-cycle, ∂F = 2 h(Cn) is smaller than 2/(n/2) = 4/n In particular, h(Cn) goes to zero as n goes to infinity

Thus we see that highly connected complete graph has a large expanding constant that grows proportionately with the number of vertices. On the other hand, the minimally connected graph has a small expanding constant that goes to zero as n increases. In this sense, h(X) does indeed give us a measure of the “quality” of the network.

That's all I know about expanders at this point. Thank you!