Topics in Probabilistic Combinatorics and Algorithms Winter, 2016
|
|
- Homer Farmer
- 5 years ago
- Views:
Transcription
1 Topics in Probabilistic Combinatorics and Algorithms Winter, Epander Graphs Review: verte boundary δ() = {v, v u }. edge boundary () = {{u, v} E, u, v / }. Remark. Of special interset are (, β) epander, which relate to the Cheeger ratios as defined below. Cheegers ratio for a subset V is defined as follows: () G: d-regular, () G: not regular, From now on, we will use definition (). Cheeger s constant: h () = δ h() = d h () = vol(δ) h() = h G = min h(). h G = min h (). Cheeger s inequality: h G λ h G.
2 Remark. Many equivalent definition for λ : < g, Lg > λ = sup g φ 0 < g, g > = sup f f()d=0 = sup f f()d=0 = sup f f()d=0 = inf sup f = inf f = inf f c < f, (D A)f > < f, Df > (φ 0 = D ) () () f (3) ()d y (f () + f (y)) (4) (f() (5) c) d ( ( f = f()d ) f() f) d d (6) (f() (7) f(y)) d d y Now, we prove Cheeger s inequality. Proof. (i) h G λ : ince λ R(f) for all f satisfies f()d = 0, where the Rayleigh quotient R(f) is defined to be R(f) = f. ()d et Then R(f) = This implies λ h G. ( f() = { + ) () + () if v if v / = h G. It remains to show (ii) λ h G. From equation (3), there eists f satisfying λ = R(f) = sup f: f()d=0 f. ()d
3 We can relabel the veritices so that f(v ) f(v ) f(v n ), and define i = {v j, j i}, j { = {v k : k j}, α = min h(i ), h( j) }. i,j uffices to show λ α since by the definition of h G, we have α h G and therefore λ h G. Let r denote the largest integer such that vol( r ) < vol( r+ ), and { f(v) f(vr ) if f(v) f(v g + (v) = r ) 0 otherwise. { f(vr ) f(v) if f(v) < f(v g (v) = r ) 0 otherwise. We consider λ = = f ()d (f() f(v r)) d ((g + () g + (y)) + (g () g (y)) ) y ( g + () + g () ) d (g +() g + (y)) ( g + () ) (use the fact: d (g +() g + (y)) (g +() + g + (y)) g +()d (g +() + g + (y)) a + b c + d min{a c, b d } w.l.o.g.) ( y g +() g +(y) ) ( ) by the Cauchy-chwarz inequality, g +()d (i = g +(v i ) g+(v i+ ) i ) ( ) by counting g +()d ( ( i g + (v i ) g+(v i+ ) ) ) α i ( ) by the definition of α g +()d (i = α g +(v i ) ( i i ) ) ) summation by parts ( g +()d = α. 3
4 Remark. G is a k-regular Ramanujan graph, Definition. λ k k σ = ma i 0 λ. Remark: G is bipartite if and only if σ =. Theorem (verte epansion). G = (V, E), V, vol ( δ) ε, (i) vol (δ) λ λ + ε. (ii) If λ ε vol (δ) ( λ + ε). Proof. Proof of (i): uppose λ R(f) = We set Then f where f satisfies ()d f()d = 0. f() = volt volt if T if T λ R(f) = T volt = f ()d ( ) T volt + volt ( volt + volt = T volt volt λ ( volt = λ volt ) λ ( ɛ) ) 4
5 Proof of (ii): Let vol (δ) ( δ) + λ ( ε)( + volδ) + λ ( ε) λ ( ε) λ ( ε) λ λ + ε. ρ = λ, f = s + ρ δ g = f c where c = f()d. Note that d g()d = 0. First, consider ince c = ( ) () f()d δ () vol( δ) f ()d By Cauchy-chwarz () ε f ()d (8) < g, (D A)g > λ R(g) = < g, Dg > < g, Ag > λ < g, Dg > we have < g, Ag > ( λ ) < g, Dg >= ρ < g, Dg >. (9) And < f, Df > = < + ρ δ, D( + ρ δ ) > = <, D > +ρ < δ, D δ > = + ρ vol(δ) < g, Dg > = < f c, D(f ) > = < f, Df > c < f, D > +c + ρ vol(δ) c (0) 5
6 We consider < f, Af > = < g + c, A(g + c) > And < g, Ag > +c ρ < g, Dg > +c by equation (9) ρ( + ρ vol(δ) c ) + c by equation (0) = ρ( + ρ vol(δ)) + ( ρ)c ρ( + ρ vol(δ)) + ( ρ)ε( + ρ vol(δ)) by equation (8) From above, we have (ρ + ε)( + ρ vol(δ)) < f, Af > e(, ) + ρe(, δ) = ( ρ)e(δ) + ρ() ρ ρ (ρ + ε)( + ρ vol(δ)) (ρ ε) (ρ + ε)ρ vol(δ) vol(δ) ρ ε (ρ + ε)ρ Theorem. G : (V, E), X, Y V, dist(x, Y ) = e(x, Y ) volxvoly volxvoly volxvoly σ σ volxvoly () Proof. ince e(x, Y ) =, A Y =, D (I L)D Y = D, (I L)D Y D X = i a i φ i a 0 = D X, D X = volx D Y = i b i φ i b 0 = D Y, D Y = voly 6
7 Then e(x, Y ) a 0 b 0 = i 0( λ i )a i b i σ a i b i i 0 i 0 ( = σ volx volx volxvolxvoly voly = σ. ) ( voly voly ) Remark. If take Y = X, the above theorem becomes ( ) volx dist(x, X) X volx σvolx volx Review: L = I D AD with eigenvalues 0 = λ 0 λ λ n and eigenvalues φ 0, φ φ n, respectively. n L = λ i φ i φ i i=0 Theorem 3 (hitting / sampling theorem). G : (V, E), undiredted, V, µ =, σ = λ. The probability that a random walk with a random stationary verte (v 0, v, v,, v t ), P r(v 0 = v) = dv = π(v) always remain in is P r[ v 0, v,, v t stay in ] (µ + σ( µ)) t. Proof. Let P = D A: transition probability matri, I s = I s, M = M. Claim: D AD λ I φ 0φ 0 I s + ( λ )I. Proof: ( ) D AD = I D AD I = I (I L)I = I φ 0φ 0 + λ i )φ i 0( i φ i I I (λ φ 0φ 0 + i ( λ )φ i φ i )I λ I φ 0φ 0 I + ( λ )I 7
8 From the claim, we have D AD λ I φ 0 +( λ ) Now, we can prove our theorem. = λ + ( λ ) (φ 0 = D ) = λ µ + σ = µ + σ( µ) P r[ v 0, v,, v t stay in ] = D I (P I ) t = D I = D ( D AI ) t ( D AD ) t D D D AD = (µ + σ( µ))t (µ + σ( µ)) t t Theorem 4 (Chernoff bound for random walks). G : (V, E), f : V [0, ], µ f = f()d. A random walk from a random stationary verte is denoted by v d 0, v,, v t, P r[ t f(vi ) µ f σ + ε ] e ε t 4 Proof. Let X i = f(v i ), X = i X i, P : transition probability matri and define T to be a diagonal matri satisfying { e rf() if = y T (, y) = 0 otherwise. Consider E(e rx ) = D T (P T )t = D ( T D T T D AD T ) t T D T D AD T t 8
9 ince D T v = d ve rf(v) v d v( + rf(v) + r ) = + rµ f + r e rµ f +r (use the fact: + e r + + if [0, ]) And T D AD T = T φ 0φ 0 + λ i )φ i 0( i φ i T T ( λ φ 0φ 0 + ( λ ) i φ i φ i ) T We have Then T D AD T λ T φ0 +( λ ) T λ ( + rµ r + r ) + σ( + r + r ) < + r(µ f + σ) + r e r(µ f +σ)+r E(e rx ) e rµ f +r e (t )(r(µ f +σ)+r ) e t(r(µ f +σ)+r ) P r[ t f(vi ) µ f σ + ε ] E(e rx ) e rt(µ f +σ+ε) e rtε+tr e ε t 4 (choose r = ε ) Applications:. Error Reduction: L BPP: L P r[ A() accepts ] 3 / L P r[ A() accepts ] 3 9
10 uppose each test takes m bit, Number of Repetitions Number of Random Bits Independent Repetitions O(k) O(km) Pairwise Independent Repetitions O( k ) O(m + k) Epander Random Walks O(k) O(m + k). ampling: f : {0, } m [0, ]. Approimate µ f within additive error ε with error probability δ. Number of amples Number of Random Bits Truly Independent O((/ε ) log(/δ)) O(m (/ε ) log(/δ)) Pairwise Independent O((/ε ) (/δ)) O(m + log(/ε) + log(/δ)) Epander Random Walk O((/ε ) log(/δ)) O(m + (/ε ) log(/δ) + log t) Eplict Construction Goal: Construct an infinite family of graphs {G i } of d regular graphs with eigenvalue λ γ. A couple of alternatives for defining eplicit constructions of epanders on a graph G = (V, E) with n vertices are: Mildy Eplict: G can be represented in time poly(log N). Fully Eplict: G can be represented in time poly(log log N). Eample 5. [0, ], =., i {0,,, b } is called a base b epansion. Known: Almost all [0, ] is normal.(in Lebesque measure) ( is normal if b in base b epansion of, each digit (0,,, b ) appear asymptotical equally often.) Open: No one can name a normal value.(e: π, e, ) Eample 6 (The problem of Ramsey numbers). The party problem, also known as the maimum clique problem, asks to find the minimum number of guests that must be invited so that at least k of them will know each other or at least k of them will not know each other. The Ramsey number R(k, k) gives the solution to the party problem: R(k, k) = min n such that every -color of K n contain monochromatic Known: 4 k R(k, k) k. K k 0
11 Construction.(discrete torus epanders). Let G = (V, E) be the graph with verte set V = Z M Z M, and edges from each node (, y) to the nodes (, y), ( +, y), (, y + ), (, + y), (y, ), where all arithmetic is modulo M. Construction.(p-cycle with inverse chords). This is the graph G = (V, E) with verte set V = Z p and edges that connect each node with the nodes: +,, and (where all arithmetic is mod p and we define 0 to be 0). Construction 3.(Ramanujan graphs). G = (V, E) is a graph with verte set V = F q { }, the finite field of prime order q s.t. q mod 4 plus one etra node representing infinity. The edges in this graph connect each node z with all z of the form: z = (a 0 + i )z + (a + ia 3 ) (a + ia 3 )z + (a 0 ia ) for a 0, a, a, a 3 Z such that a 0 + a + a + a 3 = p, a 0 is odd and positive, and a, a, a 3 are even, for some fied prime p q such that p mod 4, q is a square modulo p, and i F q such that i = mod q. Recall: s t connectivity problem: Given a graph G, does it eist a path from s to t? If we use random walks to solve it, the number of steps is approimately λ log n. Our goal is using the eplicit construction to construct a sequence of graphs {G i } such that G < G < < G k and G k has a better eigenvalues than G. Definition. G = (V, E) is a (n, d, γ)-graph if V (G) = n, maimum degree in G is d, λ γ. Now, we can define some products of graphs. Definition. quarzing: G is a (n, d, σ )-graph, where σ = λ. Tensor Product: G is a (n, d, γ)-graph and G is a (n, d, γ )-graph. Then G G = (V, E), V = {(v, v ) : v V (G), v V (G )}, E = {(v, v ) (u, u ) iff u v and u v }). Cartision Product: G G = (V, E), where V = {(v, v ) : v V (G), v V (G )} and E = {(u, u ) (v, v ) iff ((u = v, u v ) or (u = v, u v))}. Lemma 7. G is a (n, d, γ)-graph and G is a (n, d, γ )-graph, then G G is a (nn, dd, γ + γ γγ )-graph. Proof. Let F (u, u ) = f(u)g(u ), f and g are combinatorial eigenfunctions associated with λ in G and λ in G, respectively.
12 Fi u, u, F (v, v ) = f(v)g(v ) v u,v u v u,v u = f(v) g(v ) v:v u v :v u = ( λ)f(u)d u ( λ )g(u )d u = ( λ)( λ )f(u)g(u )d u d u = ( (λ + λ λλ ))f(u)g(u )d u d u F is the combinatorial eigenfunction associated with (λ + λ λλ ) in G G and the associated Laplacian eigenvalue of G G is λ + λ λλ. Definition (Zig-Zag Product Z ). Let G is a (n, d, γ )-graph and H is a (d, d, γ )-graph. Then G Z H = (V, E), V = {(u, i) : u V (G), i [d ] = V (H)}. For a, b [d ], the (a, b)th neighbor of a verte (u, i) is the verte (v, j) computed as follows: () Let i be the ath neighbor of i in H. () Let v be the i th neighbor of u in G. (3) j satisfies v j = u (4) Let j be the b th neighbor of j in H. Theorem 8. Let G is a (n, d, γ )-graph and H is a (d, d, γ )-graph. Then G Z H is a (nd, d, γ γ )-graph. Epander Construction Construction (Mildly Eplicit Epanders) Let H is a (d 4, d, 7 )-graph and define: 8 G = H G t+ = G t Z H Proposition. For all t, G t is a (d 4t, d, )-graph. Proof. By induction on t. Base case: G = H is a (d 4, d, γ ), and γ /. Induction tep: G t Z H is d4t, d, -graph. Consider G t+ : the number of vertices in G t+ is d 4t d 4 = d 4(t+) and λ(g t+ ) λ(g t ) + λ(h) (/) + (/8) = /.
13 Construction (Fully Eplicit Epanders) Let H is a (d 4, d, 7 )-graph and define: 8 G = H G t+ = (G t G t ) Z H Proposition. The number of vertices n t c t, λ t c where c is some absolute positive constant. 3
Spectral Graph Theory and its Applications
Spectral Graph Theory and its Applications Yi-Hsuan Lin Abstract This notes were given in a series of lectures by Prof Fan Chung in National Taiwan University Introduction Basic notations Let G = (V, E)
More informationLaplacian and Random Walks on Graphs
Laplacian and Random Walks on Graphs Linyuan Lu University of South Carolina Selected Topics on Spectral Graph Theory (II) Nankai University, Tianjin, May 22, 2014 Five talks Selected Topics on Spectral
More informationA generalized Alon-Boppana bound and weak Ramanujan graphs
A generalized Alon-Boppana bond and weak Ramanjan graphs Fan Chng Abstract A basic eigenvale bond de to Alon and Boppana holds only for reglar graphs. In this paper we give a generalized Alon-Boppana bond
More informationCurrent; Forest Tree Theorem; Potential Functions and their Bounds
April 13, 2008 Franklin Kenter Current; Forest Tree Theorem; Potential Functions and their Bounds 1 Introduction In this section, we will continue our discussion on current and induced current. Review
More informationA generalized Alon-Boppana bound and weak Ramanujan graphs
A generalized Alon-Boppana bond and weak Ramanjan graphs Fan Chng Department of Mathematics University of California, San Diego La Jolla, CA, U.S.A. fan@csd.ed Sbmitted: Feb 0, 206; Accepted: Jne 22, 206;
More informationTopics in Probabilistic Combinatorics and Algorithms Winter, Basic Derandomization Techniques
Topics in Probabilistic Combinatorics and Algorithms Winter, 016 3. Basic Derandomization Techniques Definition. DTIME(t(n)) : {L : L can be decided deterministically in time O(t(n)).} EXP = { L: L can
More information= 1 2 x (x 1) + 1 {x} (1 {x}). [t] dt = 1 x (x 1) + O (1), [t] dt = 1 2 x2 + O (x), (where the error is not now zero when x is an integer.
Problem Sheet,. i) Draw the graphs for [] and {}. ii) Show that for α R, α+ α [t] dt = α and α+ α {t} dt =. Hint Split these integrals at the integer which must lie in any interval of length, such as [α,
More informationACO Comprehensive Exam March 17 and 18, Computability, Complexity and Algorithms
1. Computability, Complexity and Algorithms (a) Let G(V, E) be an undirected unweighted graph. Let C V be a vertex cover of G. Argue that V \ C is an independent set of G. (b) Minimum cardinality vertex
More informationProblem Set 2. Assigned: Mon. November. 23, 2015
Pseudorandomness Prof. Salil Vadhan Problem Set 2 Assigned: Mon. November. 23, 2015 Chi-Ning Chou Index Problem Progress 1 SchwartzZippel lemma 1/1 2 Robustness of the model 1/1 3 Zero error versus 1-sided
More informationProbabilistic Proofs of Existence of Rare Events. Noga Alon
Probabilistic Proofs of Existence of Rare Events Noga Alon Department of Mathematics Sackler Faculty of Exact Sciences Tel Aviv University Ramat-Aviv, Tel Aviv 69978 ISRAEL 1. The Local Lemma In a typical
More informationLecture 13 Spectral Graph Algorithms
COMS 995-3: Advanced Algorithms March 6, 7 Lecture 3 Spectral Graph Algorithms Instructor: Alex Andoni Scribe: Srikar Varadaraj Introduction Today s topics: Finish proof from last lecture Example of random
More informationOut-colourings of Digraphs
Out-colourings of Digraphs N. Alon J. Bang-Jensen S. Bessy July 13, 2017 Abstract We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an out-colouring.
More informationZero-sum square matrices
Zero-sum square matrices Paul Balister Yair Caro Cecil Rousseau Raphael Yuster Abstract Let A be a matrix over the integers, and let p be a positive integer. A submatrix B of A is zero-sum mod p if the
More informationRandom Lifts of Graphs
27th Brazilian Math Colloquium, July 09 Plan of this talk A brief introduction to the probabilistic method. A quick review of expander graphs and their spectrum. Lifts, random lifts and their properties.
More information1 Adjacency matrix and eigenvalues
CSC 5170: Theory of Computational Complexity Lecture 7 The Chinese University of Hong Kong 1 March 2010 Our objective of study today is the random walk algorithm for deciding if two vertices in an undirected
More informationUsing Laplacian Eigenvalues and Eigenvectors in the Analysis of Frequency Assignment Problems
Using Laplacian Eigenvalues and Eigenvectors in the Analysis of Frequency Assignment Problems Jan van den Heuvel and Snežana Pejić Department of Mathematics London School of Economics Houghton Street,
More informationIntroducing the Laplacian
Introducing the Laplacian Prepared by: Steven Butler October 3, 005 A look ahead to weighted edges To this point we have looked at random walks where at each vertex we have equal probability of moving
More information6.842 Randomness and Computation March 3, Lecture 8
6.84 Randomness and Computation March 3, 04 Lecture 8 Lecturer: Ronitt Rubinfeld Scribe: Daniel Grier Useful Linear Algebra Let v = (v, v,..., v n ) be a non-zero n-dimensional row vector and P an n n
More informationHarmonic sets and the harmonic prime number theorem
Harmonic sets and the harmonic prime number theorem Version: 9th September 2004 Kevin A. Broughan and Rory J. Casey University of Waikato, Hamilton, New Zealand E-mail: kab@waikato.ac.nz We restrict primes
More informationEigenvalues, random walks and Ramanujan graphs
Eigenvalues, random walks and Ramanujan graphs David Ellis 1 The Expander Mixing lemma We have seen that a bounded-degree graph is a good edge-expander if and only if if has large spectral gap If G = (V,
More informationIsoperimetric problems
CHAPTER 2 Isoperimetric problems 2.1. History One of the earliest problems in geometry was the isoperimetric problem, which was considered by the ancient Greeks. The problem is to find, among all closed
More informationInduced subgraphs with many repeated degrees
Induced subgraphs with many repeated degrees Yair Caro Raphael Yuster arxiv:1811.071v1 [math.co] 17 Nov 018 Abstract Erdős, Fajtlowicz and Staton asked for the least integer f(k such that every graph with
More informationLecture 5: January 30
CS71 Randomness & Computation Spring 018 Instructor: Alistair Sinclair Lecture 5: January 30 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They
More informationLecture 7: ɛ-biased and almost k-wise independent spaces
Lecture 7: ɛ-biased and almost k-wise independent spaces Topics in Complexity Theory and Pseudorandomness (pring 203) Rutgers University wastik Kopparty cribes: Ben Lund, Tim Naumovitz Today we will see
More informationRandom Walk and Expander
Random Walk and Expander We have seen some major results based on the assumption that certain random/hard objects exist. We will see what we can achieve unconditionally using explicit constructions of
More informationEconomics 205 Exercises
Economics 05 Eercises Prof. Watson, Fall 006 (Includes eaminations through Fall 003) Part 1: Basic Analysis 1. Using ε and δ, write in formal terms the meaning of lim a f() = c, where f : R R.. Write the
More informationNumber Theory, Algebra and Analysis. William Yslas Vélez Department of Mathematics University of Arizona
Number Theory, Algebra and Analysis William Yslas Vélez Department of Mathematics University of Arizona O F denotes the ring of integers in the field F, it mimics Z in Q How do primes factor as you consider
More informationLecture Introduction. 2 Brief Recap of Lecture 10. CS-621 Theory Gems October 24, 2012
CS-62 Theory Gems October 24, 202 Lecture Lecturer: Aleksander Mądry Scribes: Carsten Moldenhauer and Robin Scheibler Introduction In Lecture 0, we introduced a fundamental object of spectral graph theory:
More informationThe Pigeonhole Principle
The Pigeonhole Principle 2 2.1 The Pigeonhole Principle The pigeonhole principle is one of the most used tools in combinatorics, and one of the simplest ones. It is applied frequently in graph theory,
More information1 Take-home exam and final exam study guide
Math 215 - Introduction to Advanced Mathematics Fall 2013 1 Take-home exam and final exam study guide 1.1 Problems The following are some problems, some of which will appear on the final exam. 1.1.1 Number
More informationPGSS Discrete Math Solutions to Problem Set #4. Note: signifies the end of a problem, and signifies the end of a proof.
PGSS Discrete Math Solutions to Problem Set #4 Note: signifies the end of a problem, and signifies the end of a proof. 1. Prove that for any k N, there are k consecutive composite numbers. (Hint: (k +
More informationExpander and Derandomization
Expander and Derandomization Many derandomization results are based on the assumption that certain random/hard objects exist. Some unconditional derandomization can be achieved using explicit constructions
More information2 Generating Functions
2 Generating Functions In this part of the course, we re going to introduce algebraic methods for counting and proving combinatorial identities. This is often greatly advantageous over the method of finding
More informationLecture 14: Random Walks, Local Graph Clustering, Linear Programming
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 14: Random Walks, Local Graph Clustering, Linear Programming Lecturer: Shayan Oveis Gharan 3/01/17 Scribe: Laura Vonessen Disclaimer: These
More information: Error Correcting Codes. December 2017 Lecture 10
0368307: Error Correcting Codes. December 017 Lecture 10 The Linear-Programming Bound Amnon Ta-Shma and Dean Doron 1 The LP-bound We will prove the Linear-Programming bound due to [, 1], which gives an
More informationTHE RAINBOW DOMINATION NUMBER OF A DIGRAPH
Kragujevac Journal of Mathematics Volume 37() (013), Pages 57 68. THE RAINBOW DOMINATION NUMBER OF A DIGRAPH J. AMJADI 1, A. BAHREMANDPOUR 1, S. M. SHEIKHOLESLAMI 1, AND L. VOLKMANN Abstract. Let D = (V,
More informationStanford University CS366: Graph Partitioning and Expanders Handout 13 Luca Trevisan March 4, 2013
Stanford University CS366: Graph Partitioning and Expanders Handout 13 Luca Trevisan March 4, 2013 Lecture 13 In which we construct a family of expander graphs. The problem of constructing expander graphs
More informationReading group: proof of the PCP theorem
Reading group: proof of the PCP theorem 1 The PCP Theorem The usual formulation of the PCP theorem is equivalent to: Theorem 1. There exists a finite constraint language A such that the following problem
More information1 Random Walks and Electrical Networks
CME 305: Discrete Mathematics and Algorithms Random Walks and Electrical Networks Random walks are widely used tools in algorithm design and probabilistic analysis and they have numerous applications.
More informationSmall gaps between primes
CRM, Université de Montréal Princeton/IAS Number Theory Seminar March 2014 Introduction Question What is lim inf n (p n+1 p n )? In particular, is it finite? Introduction Question What is lim inf n (p
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More informationLecture 15: Expanders
CS 710: Complexity Theory 10/7/011 Lecture 15: Expanders Instructor: Dieter van Melkebeek Scribe: Li-Hsiang Kuo In the last lecture we introduced randomized computation in terms of machines that have access
More informationSpectral Graph Theory
Spectral raph Theory to appear in Handbook of Linear Algebra, second edition, CCR Press Steve Butler Fan Chung There are many different ways to associate a matrix with a graph (an introduction of which
More informationOn the Subsequence of Primes Having Prime Subscripts
3 47 6 3 Journal of Integer Sequences, Vol. (009, Article 09..3 On the Subsequence of Primes Having Prime Subscripts Kevin A. Broughan and A. Ross Barnett University of Waikato Hamilton, New Zealand kab@waikato.ac.nz
More informationRoth s Theorem on 3-term Arithmetic Progressions
Roth s Theorem on 3-term Arithmetic Progressions Mustazee Rahman 1 Introduction This article is a discussion about the proof of a classical theorem of Roth s regarding the existence of three term arithmetic
More informationSYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS
SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..
More informationSum and shifted-product subsets of product-sets over finite rings
Sum and shifted-product subsets of product-sets over finite rings Le Anh Vinh University of Education Vietnam National University, Hanoi vinhla@vnu.edu.vn Submitted: Jan 6, 2012; Accepted: May 25, 2012;
More informationProbabilistic Combinatorics. Jeong Han Kim
Probabilistic Combinatorics Jeong Han Kim 1 Tentative Plans 1. Basics for Probability theory Coin Tossing, Expectation, Linearity of Expectation, Probability vs. Expectation, Bool s Inequality 2. First
More informationMa/CS 6b Class 15: The Probabilistic Method 2
Ma/CS 6b Class 15: The Probabilistic Method 2 By Adam Sheffer Reminder: Random Variables A random variable is a function from the set of possible events to R. Example. Say that we flip five coins. We can
More informationDiscrete Mathematics
Discrete Mathematics Workshop Organized by: ACM Unit, ISI Tutorial-1 Date: 05.07.2017 (Q1) Given seven points in a triangle of unit area, prove that three of them form a triangle of area not exceeding
More informationRamsey Unsaturated and Saturated Graphs
Ramsey Unsaturated and Saturated Graphs P Balister J Lehel RH Schelp March 20, 2005 Abstract A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number,
More informationThe concentration of the chromatic number of random graphs
The concentration of the chromatic number of random graphs Noga Alon Michael Krivelevich Abstract We prove that for every constant δ > 0 the chromatic number of the random graph G(n, p) with p = n 1/2
More informationList coloring hypergraphs
List coloring hypergraphs Penny Haxell Jacques Verstraete Department of Combinatorics and Optimization University of Waterloo Waterloo, Ontario, Canada pehaxell@uwaterloo.ca Department of Mathematics University
More informationTHE CONVEX HULL OF THE PRIME NUMBER GRAPH
THE CONVEX HULL OF THE PRIME NUMBER GRAPH NATHAN MCNEW Abstract Let p n denote the n-th prime number, and consider the prime number graph, the collection of points n, p n in the plane Pomerance uses the
More information2-Distance Problems. Combinatorics, 2016 Fall, USTC Week 16, Dec 20&22. Theorem 1. (Frankl-Wilson, 1981) If F is an L-intersecting family in 2 [n],
Combinatorics, 206 Fall, USTC Week 6, Dec 20&22 2-Distance Problems Theorem (Frankl-Wilson, 98 If F is an L-intersecting family in 2 [n], then F L k=0 ( n k Proof Let F = {A, A 2,, A m } where A A 2 A
More informationList of Theorems. Mat 416, Introduction to Graph Theory. Theorem 1 The numbers R(p, q) exist and for p, q 2,
List of Theorems Mat 416, Introduction to Graph Theory 1. Ramsey s Theorem for graphs 8.3.11. Theorem 1 The numbers R(p, q) exist and for p, q 2, R(p, q) R(p 1, q) + R(p, q 1). If both summands on the
More informationNowhere zero flow. Definition: A flow on a graph G = (V, E) is a pair (D, f) such that. 1. D is an orientation of G. 2. f is a function on E.
Nowhere zero flow Definition: A flow on a graph G = (V, E) is a pair (D, f) such that 1. D is an orientation of G. 2. f is a function on E. 3. u N + D (v) f(uv) = w ND f(vw) for every (v) v V. Example:
More informationF 99 Final Quiz Solutions
Massachusetts Institute of Technology Handout 53 6.04J/18.06J: Mathematics for Computer Science May 11, 000 Professors David Karger and Nancy Lynch F 99 Final Quiz Solutions Problem 1 [18 points] Injections
More informationThe PCP Theorem by Gap Amplification
Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 46 (2005) The PCP Theorem by Gap Amplification Irit Dinur September 25, 2005 Abstract We describe a new proof of the PCP theorem
More informationACO Comprehensive Exam March 20 and 21, Computability, Complexity and Algorithms
1. Computability, Complexity and Algorithms Part a: You are given a graph G = (V,E) with edge weights w(e) > 0 for e E. You are also given a minimum cost spanning tree (MST) T. For one particular edge
More information< k 2n. 2 1 (n 2). + (1 p) s) N (n < 1
List of Problems jacques@ucsd.edu Those question with a star next to them are considered slightly more challenging. Problems 9, 11, and 19 from the book The probabilistic method, by Alon and Spencer. Question
More informationC.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series
C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also
More informationThe Simplest Construction of Expanders
Spectral Graph Theory Lecture 14 The Simplest Construction of Expanders Daniel A. Spielman October 16, 2009 14.1 Overview I am going to present the simplest construction of expanders that I have been able
More information1 T 1 = where 1 is the all-ones vector. For the upper bound, let v 1 be the eigenvector corresponding. u:(u,v) E v 1(u)
CME 305: Discrete Mathematics and Algorithms Instructor: Reza Zadeh (rezab@stanford.edu) Final Review Session 03/20/17 1. Let G = (V, E) be an unweighted, undirected graph. Let λ 1 be the maximum eigenvalue
More informationAlgebraic Constructions of Graphs
Spectral Graph Theory Lecture 15 Algebraic Constructions of Graphs Daniel A. Spielman October 17, 2012 15.1 Overview In this lecture, I will explain how to make graphs from linear error-correcting codes.
More informationGRAPHS CONTAINING TRIANGLES ARE NOT 3-COMMON
GRAPHS CONTAINING TRIANGLES ARE NOT 3-COMMON JAMES CUMMINGS AND MICHAEL YOUNG Abstract. A finite graph G is k-common if the minimum (over all k- colourings of the edges of K n) of the number of monochromatic
More informationUse mathematical induction in Exercises 3 17 to prove summation formulae. Be sure to identify where you use the inductive hypothesis.
Exercises Exercises 1. There are infinitely many stations on a train route. Suppose that the train stops at the first station and suppose that if the train stops at a station, then it stops at the next
More informationBounds for pairs in partitions of graphs
Bounds for pairs in partitions of graphs Jie Ma Xingxing Yu School of Mathematics Georgia Institute of Technology Atlanta, GA 30332-0160, USA Abstract In this paper we study the following problem of Bollobás
More informationThe Probabilistic Method
The Probabilistic Method In Graph Theory Ehssan Khanmohammadi Department of Mathematics The Pennsylvania State University February 25, 2010 What do we mean by the probabilistic method? Why use this method?
More informationProperties of Ramanujan Graphs
Properties of Ramanujan Graphs Andrew Droll 1, 1 Department of Mathematics and Statistics, Jeffery Hall, Queen s University Kingston, Ontario, Canada Student Number: 5638101 Defense: 27 August, 2008, Wed.
More informationMATH 61-02: PRACTICE PROBLEMS FOR FINAL EXAM
MATH 61-02: PRACTICE PROBLEMS FOR FINAL EXAM (FP1) The exclusive or operation, denoted by and sometimes known as XOR, is defined so that P Q is true iff P is true or Q is true, but not both. Prove (through
More informationEigenvalues of graphs
Eigenvalues of graphs F. R. K. Chung University of Pennsylvania Philadelphia PA 19104 September 11, 1996 1 Introduction The study of eigenvalues of graphs has a long history. From the early days, representation
More informationLaplacians of Graphs, Spectra and Laplacian polynomials
Laplacians of Graphs, Spectra and Laplacian polynomials Lector: Alexander Mednykh Sobolev Institute of Mathematics Novosibirsk State University Winter School in Harmonic Functions on Graphs and Combinatorial
More informationWeek 15-16: Combinatorial Design
Week 15-16: Combinatorial Design May 8, 2017 A combinatorial design, or simply a design, is an arrangement of the objects of a set into subsets satisfying certain prescribed properties. The area of combinatorial
More informationRamsey Theory. May 24, 2015
Ramsey Theory May 24, 2015 1 König s Lemma König s Lemma is a basic tool to move between finite and infinite combinatorics. To be concise, we use the notation [k] = {1, 2,..., k}, and [X] r will denote
More informationTaylor Series and Series Convergence (Online)
7in 0in Felder c02_online.te V3 - February 9, 205 9:5 A.M. Page CHAPTER 2 Taylor Series and Series Convergence (Online) 2.8 Asymptotic Epansions In introductory calculus classes the statement this series
More informationMA2223 Tutorial solutions Part 1. Metric spaces
MA2223 Tutorial solutions Part 1. Metric spaces T1 1. Show that the function d(,y) = y defines a metric on R. The given function is symmetric and non-negative with d(,y) = 0 if and only if = y. It remains
More informationPaul Erdős and Graph Ramsey Theory
Paul Erdős and Graph Ramsey Theory Benny Sudakov ETH and UCLA Ramsey theorem Ramsey theorem Definition: The Ramsey number r(s, n) is the minimum N such that every red-blue coloring of the edges of a complete
More informationDecomposing oriented graphs into transitive tournaments
Decomposing oriented graphs into transitive tournaments Raphael Yuster Department of Mathematics University of Haifa Haifa 39105, Israel Abstract For an oriented graph G with n vertices, let f(g) denote
More information= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2
8. p-adic numbers 8.1. Motivation: Solving x 2 a (mod p n ). Take an odd prime p, and ( an) integer a coprime to p. Then, as we know, x 2 a (mod p) has a solution x Z iff = 1. In this case we can suppose
More informationFlat primes and thin primes
Flat primes and thin primes Kevin A. Broughan and Zhou Qizhi University of Waikato, Hamilton, New Zealand Version: 0th October 2008 E-mail: kab@waikato.ac.nz, qz49@waikato.ac.nz Flat primes and thin primes
More informationThe chromatic number of random Cayley graphs
The chromatic number of random Cayley graphs Noga Alon Abstract We consider the typical behaviour of the chromatic number of a random Cayley graph of a given group of order n with respect to a randomly
More informationTaylor Series and Asymptotic Expansions
Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.
More informationThe Removal Lemma for Tournaments
The Removal Lemma for Tournaments Jacob Fox Lior Gishboliner Asaf Shapira Raphael Yuster August 8, 2017 Abstract Suppose one needs to change the direction of at least ɛn 2 edges of an n-vertex tournament
More informationWAITING FOR A BAT TO FLY BY (IN POLYNOMIAL TIME)
WAITING FOR A BAT TO FLY BY (IN POLYNOMIAL TIME ITAI BENJAMINI, GADY KOZMA, LÁSZLÓ LOVÁSZ, DAN ROMIK, AND GÁBOR TARDOS Abstract. We observe returns of a simple random wal on a finite graph to a fixed node,
More informationOn certain integral Schreier graphs of the symmetric group
On certain integral Schreier graphs of the symmetric group Paul E. Gunnells Department of Mathematics and Statistics University of Massachusetts Amherst, Massachusetts, USA gunnells@math.umass.edu Richard
More information9 - The Combinatorial Nullstellensatz
9 - The Combinatorial Nullstellensatz Jacques Verstraëte jacques@ucsd.edu Hilbert s nullstellensatz says that if F is an algebraically closed field and f and g 1, g 2,..., g m are polynomials in F[x 1,
More informationLecture 5: Rules of Differentiation. First Order Derivatives
Lecture 5: Rules of Differentiation First order derivatives Higher order derivatives Partial differentiation Higher order partials Differentials Derivatives of implicit functions Generalized implicit function
More informationMax-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig
Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig On the spectrum of the normalized Laplacian for signed graphs: Interlacing, contraction, and replication by Fatihcan M. Atay and Hande
More informationCycle lengths in sparse graphs
Cycle lengths in sparse graphs Benny Sudakov Jacques Verstraëte Abstract Let C(G) denote the set of lengths of cycles in a graph G. In the first part of this paper, we study the minimum possible value
More informationOn splitting digraphs
On splitting digraphs arxiv:707.03600v [math.co] 0 Apr 08 Donglei Yang a,, Yandong Bai b,, Guanghui Wang a,, Jianliang Wu a, a School of Mathematics, Shandong University, Jinan, 5000, P. R. China b Department
More informationInduced Graph Ramsey Theory
Induced Graph Ramsey Theory Marcus Schaefer School of CTI DePaul University 243 South Wabash Avenue Chicago, Illinois 60604, USA schaefer@csdepauledu June 28, 2000 Pradyut Shah Department of Computer Science
More informationMath 104: Homework 7 solutions
Math 04: Homework 7 solutions. (a) The derivative of f () = is f () = 2 which is unbounded as 0. Since f () is continuous on [0, ], it is uniformly continous on this interval by Theorem 9.2. Hence for
More informationGRAPH EMBEDDING LECTURE NOTE
GRAPH EMBEDDING LECTURE NOTE JAEHOON KIM Abstract. In this lecture, we aim to learn several techniques to find sufficient conditions on a dense graph G to contain a sparse graph H as a subgraph. In particular,
More information(1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define
Homework, Real Analysis I, Fall, 2010. (1) Consider the space S consisting of all continuous real-valued functions on the closed interval [0, 1]. For f, g S, define ρ(f, g) = 1 0 f(x) g(x) dx. Show that
More informationExpansion and Isoperimetric Constants for Product Graphs
Expansion and Isoperimetric Constants for Product Graphs C. Houdré and T. Stoyanov May 4, 2004 Abstract Vertex and edge isoperimetric constants of graphs are studied. Using a functional-analytic approach,
More informationMonochromatic and Rainbow Colorings
Chapter 11 Monochromatic and Rainbow Colorings There are instances in which we will be interested in edge colorings of graphs that do not require adjacent edges to be assigned distinct colors Of course,
More informationCSE 1400 Applied Discrete Mathematics Proofs
CSE 1400 Applied Discrete Mathematics Proofs Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Axioms 1 Logical Axioms 2 Models 2 Number Theory 3 Graph Theory 4 Set Theory 4
More information10.4 The Kruskal Katona theorem
104 The Krusal Katona theorem 141 Example 1013 (Maximum weight traveling salesman problem We are given a complete directed graph with non-negative weights on edges, and we must find a maximum weight Hamiltonian
More informationMathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations
Mathematics Course 111: Algebra I Part I: Algebraic Structures, Sets and Permutations D. R. Wilkins Academic Year 1996-7 1 Number Systems and Matrix Algebra Integers The whole numbers 0, ±1, ±2, ±3, ±4,...
More information