Peter J. Dukes. 22 August, 2012
|
|
- Justin Arnold
- 6 years ago
- Views:
Transcription
1 22 August, 22
2
3 Graph decomposition Let G and H be graphs on m n vertices. A decompostion of G into copies of H is a collection {H i } of subgraphs of G such that each H i = H, and every edge of G belongs to exactly one H i. When G is complete, these objects are also known as H-designs.
4 Graph decomposition Let G and H be graphs on m n vertices. A decompostion of G into copies of H is a collection {H i } of subgraphs of G such that each H i = H, and every edge of G belongs to exactly one H i. When G is complete, these objects are also known as H-designs.
5 Example. Let G and H be the graphs shown below. G H
6 Example. Let G and H be the graphs shown below. G H
7 decomposition Relaxation: a fractional decomposition of G into H is a collection {(H i, w i )}, where w i is a positive real (w.l.o.g. rational) weight assigned to subgraph H i = H of G such that for every edge e of G, i : H i has e w i = 1. If there exists a fractional decomposition of G into H, let s write H Q G.
8 Example. Let P 2 be the path on two edges. Then P 2 Q C 5 by taking each of the 5 embeddings of P 2 in C 5 with weight 1 2. Example. Let H have n vertices. It is clear that H Q K n : simply take all copies of H on the points of G with suitable (constant) weight. By contrast, H Q G may sometimes fail; for instance, H G is necessary. In fact, we need the existsence of many subgraphs of H in G.
9 Example. Let P 2 be the path on two edges. Then P 2 Q C 5 by taking each of the 5 embeddings of P 2 in C 5 with weight 1 2. Example. Let H have n vertices. It is clear that H Q K n : simply take all copies of H on the points of G with suitable (constant) weight. By contrast, H Q G may sometimes fail; for instance, H G is necessary. In fact, we need the existsence of many subgraphs of H in G.
10 Example. Let P 2 be the path on two edges. Then P 2 Q C 5 by taking each of the 5 embeddings of P 2 in C 5 with weight 1 2. Example. Let H have n vertices. It is clear that H Q K n : simply take all copies of H on the points of G with suitable (constant) weight. By contrast, H Q G may sometimes fail; for instance, H G is necessary. In fact, we need the existsence of many subgraphs of H in G.
11 Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 :
12 Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 :
13 Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 :
14 Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 :
15 Non-Example. C 5 obviously does not admit a fractional decomposition into K 3, even though there is a signed weighting of triangles on 5 points which yields C 5 :
16 Adjacency matrices Let G, H have adjacency matrices A G, A H, respectively. Then G Q H is equivalent to a decomposition A G = i w i Q i A H Q i, where each Q i is an m n (0, 1)-matrix having exactly one 1 per column, and the w i are positive rationals.
17 We can extend all these ideas to t-uniform hypergraphs, or t-graphs for short. In what follows, think of a fixed, small t-graph H. Question. How dense (say in terms of (t 1)-subset degrees) does a large t-graph G need to be in order to guarantee H Q G? By transitivity of Q, it is enough to answer this question for complete t-graphs H = K t k. But this differs from a t-design in two respects. The decompositions here are fractional. Only a dense collection of prescribed edges is to be covered.
18 We can extend all these ideas to t-uniform hypergraphs, or t-graphs for short. In what follows, think of a fixed, small t-graph H. Question. How dense (say in terms of (t 1)-subset degrees) does a large t-graph G need to be in order to guarantee H Q G? By transitivity of Q, it is enough to answer this question for complete t-graphs H = K t k. But this differs from a t-design in two respects. The decompositions here are fractional. Only a dense collection of prescribed edges is to be covered.
19 We can extend all these ideas to t-uniform hypergraphs, or t-graphs for short. In what follows, think of a fixed, small t-graph H. Question. How dense (say in terms of (t 1)-subset degrees) does a large t-graph G need to be in order to guarantee H Q G? By transitivity of Q, it is enough to answer this question for complete t-graphs H = K t k. But this differs from a t-design in two respects. The decompositions here are fractional. Only a dense collection of prescribed edges is to be covered.
20 Main Theorem. For integers k t 2, there exists v 0 (t, k) and C = C(t) such that, for v > v 0 and ɛ < Ck 2t, any (1 ɛ)-dense t-graph G on v vertices admits a rational decomposition into copies of K t k. Proof idea. Estimate perturbations of the Johnson scheme s eigenvalues. From this, show that a certain square linear system has nonnegative solutions for small ɛ, by Cramer s rule. Interpret such solutions as fractional decompositions.
21 Main Theorem. For integers k t 2, there exists v 0 (t, k) and C = C(t) such that, for v > v 0 and ɛ < Ck 2t, any (1 ɛ)-dense t-graph G on v vertices admits a rational decomposition into copies of K t k. Proof idea. Estimate perturbations of the Johnson scheme s eigenvalues. From this, show that a certain square linear system has nonnegative solutions for small ɛ, by Cramer s rule. Interpret such solutions as fractional decompositions.
22 Set up Let V be a v-set and put X = ( V k ). For a set system F X, write F(T ) for the number of times T is covered by F. For fixed U ( ) V ( t, consider the family F = X [U] of all v t k t) k-subsets of V which contain U. Then ( ) v T U X [U](T ) =, k T U since this counts the number of k-subsets containing both T and U.
23 Set up Let V be a v-set and put X = ( V k ). For a set system F X, write F(T ) for the number of times T is covered by F. For fixed U ( ) V ( t, consider the family F = X [U] of all v t k t) k-subsets of V which contain U. Then ( ) v T U X [U](T ) =, k T U since this counts the number of k-subsets containing both T and U.
24 Set up Let V be a v-set and put X = ( V k ). For a set system F X, write F(T ) for the number of times T is covered by F. For fixed U ( ) V ( t, consider the family F = X [U] of all v t k t) k-subsets of V which contain U. Then ( ) v T U X [U](T ) =, k T U since this counts the number of k-subsets containing both T and U.
25 Therefore, where ξ i = for i = 0, 1,..., t. X [U](T ) = ξ T \U, ( ) v t i = v k t i k t i (k t i)! + o(v k t i ). Let n = ( v t). Define the n n matrix M by for T, U ( V t ). M(T, U) = ξ T \U = X [U](T ),
26 Therefore, where ξ i = for i = 0, 1,..., t. X [U](T ) = ξ T \U, ( ) v t i = v k t i k t i (k t i)! + o(v k t i ). Let n = ( v t). Define the n n matrix M by for T, U ( V t ). M(T, U) = ξ T \U = X [U](T ),
27 In fact, M factors as M = WW, where W is the well-known inclusion matrix of t-subsets versus k-subsets. Note that M = M and the constant column (row) sum of M is ( )( ) v t k. k t t
28 In fact, M factors as M = WW, where W is the well-known inclusion matrix of t-subsets versus k-subsets. Note that M = M and the constant column (row) sum of M is ( )( ) v t k. k t t
29 Note that a nonnegative solution x to Mx = 1 induces a fractional decomposition K t k Q K t v. Simply take each X [U] with weight x(u), and the total coverage at T is x(u)m(t, U) = (Mx)(T ) = 1 U on each t-set T.
30 Restriction to edges of G But decomposing a non-complete t-graph G is not so easy. We must restrict our attention to k-subsets that cover only those edges present in G. To this end, define X G as the family of all k-subsets which induce a clique in G. In other words, K X G if and only if K V with K = k, and T K with T = t implies T is an edge of G. Now consider X G [U], the family of all k-subsets on V which contain U and also induce a clique in G. Define the G G matrix M, with rows and columns indexed by edges of G, by M(T, U) = X G [U](T ).
31 Restriction to edges of G But decomposing a non-complete t-graph G is not so easy. We must restrict our attention to k-subsets that cover only those edges present in G. To this end, define X G as the family of all k-subsets which induce a clique in G. In other words, K X G if and only if K V with K = k, and T K with T = t implies T is an edge of G. Now consider X G [U], the family of all k-subsets on V which contain U and also induce a clique in G. Define the G G matrix M, with rows and columns indexed by edges of G, by M(T, U) = X G [U](T ).
32 Restriction to edges of G But decomposing a non-complete t-graph G is not so easy. We must restrict our attention to k-subsets that cover only those edges present in G. To this end, define X G as the family of all k-subsets which induce a clique in G. In other words, K X G if and only if K V with K = k, and T K with T = t implies T is an edge of G. Now consider X G [U], the family of all k-subsets on V which contain U and also induce a clique in G. Define the G G matrix M, with rows and columns indexed by edges of G, by M(T, U) = X G [U](T ).
33 Note M is symmetric, and it is approximately a principal submatrix of M in the positions indexed by G. And, most importantly, a nonnegative solution x to Mx = 1 if it exists, yields a fractional decomposition K t k Q G.
34 Counting the missing cliques Lemma. Suppose G is a (1 ɛ)-dense simple t-graph. 1. Given an edge T and i with 0 i t, there are at least ( )( ) [ ( ) ] t v t + i 1 ɛ + o(1) i i i edges U such that T \ U = i and T U induces a clique in G. 2. If T and U are edges of G with T \ U = i and such that T U induces a clique in G, then there are at least ( ] t + i ( v t i k t i ) [ 1 (( k t ) i )) ɛ + o(1) k-subsets containing T U and inducing a clique in G. Proof: easy counting Remarks: Part (a) essentially asserts that most entries of M are nonzero, while part (b) asserts that those nonzero entries are close to those of M.
35 Counting the missing cliques Lemma. Suppose G is a (1 ɛ)-dense simple t-graph. 1. Given an edge T and i with 0 i t, there are at least ( )( ) [ ( ) ] t v t + i 1 ɛ + o(1) i i i edges U such that T \ U = i and T U induces a clique in G. 2. If T and U are edges of G with T \ U = i and such that T U induces a clique in G, then there are at least ( ] t + i ( v t i k t i ) [ 1 (( k t ) i )) ɛ + o(1) k-subsets containing T U and inducing a clique in G. Proof: easy counting Remarks: Part (a) essentially asserts that most entries of M are nonzero, while part (b) asserts that those nonzero entries are close to those of M.
36 Eigenvalues The eigenvalues of M are easy to get explicitly from Delsarte theory. Since M = t i=0 ξ ia i, it follows that M = t j=0 θ je j, where θ j = t ξ i P ij i=0... = ( )( ) k j v t + o(v k t ). t j k t Recall that the E i are orthogonal idempotents. So these coefficients θ i are the eigenvalues of M.
37 Perturbation Lemma. Suppose A and A are Hermitian matrices such that every eigenvalue of A is greater than A. Then A + A is positive definite. Let s invoke this with A = M G and A = M := M M G. We know that every eigenvalue of A is at least θ t The earlier lemma offers an estimate A < θ t ( k t ) 2 ɛ + o(v k t ).
38 Cramer s rule Finally, for Cramer s rule, we need to estimate eigenvalues of the matrix obtained by replacing a column of M by 1. This is accomplished by interlacing results, which guarantee a lower bound of 1 2 θ t. Conclusion: Our solution vector M 1 1 is (asymptotically in v) entrywise positive for ɛ < 1 2 ( ) k 2. t
39 Cramer s rule Finally, for Cramer s rule, we need to estimate eigenvalues of the matrix obtained by replacing a column of M by 1. This is accomplished by interlacing results, which guarantee a lower bound of 1 2 θ t. Conclusion: Our solution vector M 1 1 is (asymptotically in v) entrywise positive for ɛ < 1 2 ( ) k 2. t
40 An application decomposition of certain high-degree circulants has proved useful in balanced sampling plans. Here, one wishes to fairly sample, by k-subsets, all pairs of elements which are not too close together. The rational weights w i associated with each sample H i = Kk is the probability with which that sample is selected for testing. The total probability with which a pair of points is tested is either zero or one, according to the definition of close. So fractional decompositions are useful in statistics.
41 P. Dukes, Rational decomposition of dense hypergraphs and some related eigenvalue estimates, LAA, 22. P. Dukes and A.C.H. Ling, Existence of balanced sampling plans avoiding cyclic distances, Metrika, C.D. Godsil, Notes on Association Schemes. University of Waterloo, R. Yuster, decompositions of dense hypergraphs. Approximation, randomization and combinatorial optimization, THE END -
The doubly negative matrix completion problem
The doubly negative matrix completion problem C Mendes Araújo, Juan R Torregrosa and Ana M Urbano CMAT - Centro de Matemática / Dpto de Matemática Aplicada Universidade do Minho / Universidad Politécnica
More informationCentral Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J
Central Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J Frank Curtis, John Drew, Chi-Kwong Li, and Daniel Pragel September 25, 2003 Abstract We study central groupoids, central
More informationIntriguing sets of vertices of regular graphs
Intriguing sets of vertices of regular graphs Bart De Bruyn and Hiroshi Suzuki February 18, 2010 Abstract Intriguing and tight sets of vertices of point-line geometries have recently been studied in the
More informationAverage Mixing Matrix of Trees
Electronic Journal of Linear Algebra Volume 34 Volume 34 08 Article 9 08 Average Mixing Matrix of Trees Chris Godsil University of Waterloo, cgodsil@uwaterloo.ca Krystal Guo Université libre de Bruxelles,
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 information5 Quiver Representations
5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )
More informationChapter 2 Spectra of Finite Graphs
Chapter 2 Spectra of Finite Graphs 2.1 Characteristic Polynomials Let G = (V, E) be a finite graph on n = V vertices. Numbering the vertices, we write down its adjacency matrix in an explicit form of n
More informationA lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo
A lower bound for the Laplacian eigenvalues of a graph proof of a conjecture by Guo A. E. Brouwer & W. H. Haemers 2008-02-28 Abstract We show that if µ j is the j-th largest Laplacian eigenvalue, and d
More informationRATIONAL REALIZATION OF MAXIMUM EIGENVALUE MULTIPLICITY OF SYMMETRIC TREE SIGN PATTERNS. February 6, 2006
RATIONAL REALIZATION OF MAXIMUM EIGENVALUE MULTIPLICITY OF SYMMETRIC TREE SIGN PATTERNS ATOSHI CHOWDHURY, LESLIE HOGBEN, JUDE MELANCON, AND RANA MIKKELSON February 6, 006 Abstract. A sign pattern is a
More informationRelationships between the Completion Problems for Various Classes of Matrices
Relationships between the Completion Problems for Various Classes of Matrices Leslie Hogben* 1 Introduction A partial matrix is a matrix in which some entries are specified and others are not (all entries
More informationThe Maximum Likelihood Threshold of a Graph
The Maximum Likelihood Threshold of a Graph Elizabeth Gross and Seth Sullivant San Jose State University, North Carolina State University August 28, 2014 Seth Sullivant (NCSU) Maximum Likelihood Threshold
More informationNew feasibility conditions for directed strongly regular graphs
New feasibility conditions for directed strongly regular graphs Sylvia A. Hobart Jason Williford Department of Mathematics University of Wyoming Laramie, Wyoming, U.S.A sahobart@uwyo.edu, jwillif1@uwyo.edu
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 informationThe minimum rank of matrices and the equivalence class graph
Linear Algebra and its Applications 427 (2007) 161 170 wwwelseviercom/locate/laa The minimum rank of matrices and the equivalence class graph Rosário Fernandes, Cecília Perdigão Departamento de Matemática,
More informationPCA with random noise. Van Ha Vu. Department of Mathematics Yale University
PCA with random noise Van Ha Vu Department of Mathematics Yale University An important problem that appears in various areas of applied mathematics (in particular statistics, computer science and numerical
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 informationA new graph parameter related to bounded rank positive semidefinite matrix completions
Mathematical Programming manuscript No. (will be inserted by the editor) Monique Laurent Antonios Varvitsiotis A new graph parameter related to bounded rank positive semidefinite matrix completions the
More informationPacking and Covering Dense Graphs
Packing and Covering Dense Graphs Noga Alon Yair Caro Raphael Yuster Abstract Let d be a positive integer. A graph G is called d-divisible if d divides the degree of each vertex of G. G is called nowhere
More information1 The linear algebra of linear programs (March 15 and 22, 2015)
1 The linear algebra of linear programs (March 15 and 22, 2015) Many optimization problems can be formulated as linear programs. The main features of a linear program are the following: Variables are real
More informationFiedler s Theorems on Nodal Domains
Spectral Graph Theory Lecture 7 Fiedler s Theorems on Nodal Domains Daniel A. Spielman September 19, 2018 7.1 Overview In today s lecture we will justify some of the behavior we observed when using eigenvectors
More informationOn star forest ascending subgraph decomposition
On star forest ascending subgraph decomposition Josep M. Aroca and Anna Lladó Department of Mathematics, Univ. Politècnica de Catalunya Barcelona, Spain josep.m.aroca@upc.edu,aina.llado@upc.edu Submitted:
More informationNOTE ON CYCLIC DECOMPOSITIONS OF COMPLETE BIPARTITE GRAPHS INTO CUBES
Discussiones Mathematicae Graph Theory 19 (1999 ) 219 227 NOTE ON CYCLIC DECOMPOSITIONS OF COMPLETE BIPARTITE GRAPHS INTO CUBES Dalibor Fronček Department of Applied Mathematics Technical University Ostrava
More informationMath 408 Advanced Linear Algebra
Math 408 Advanced Linear Algebra Chi-Kwong Li Chapter 4 Hermitian and symmetric matrices Basic properties Theorem Let A M n. The following are equivalent. Remark (a) A is Hermitian, i.e., A = A. (b) x
More informationSome parametric inequalities on strongly regular graphs spectra
INTERNATIONAL JOURNAL OF PURE MATHEMATICS Volume 3, 06 Some parametric inequalities on strongly regular graphs spectra Vasco Moço Mano and Luís de Almeida Vieira Email: vascomocomano@gmailcom Faculty of
More information2.57 PART E: THE FUNDAMENTAL THEOREM OF ALGEBRA (FTA) The Fundamental Theorem of Algebra (FTA)
2.57 PART E: THE FUNDAMENTAL THEOREM OF ALGEBRA (FTA) The Fundamental Theorem of Algebra (FTA) If f ( x) is a nonconstant n th -degree polynomial in standard form with real coefficients, then it must have
More informationInequalities for matrices and the L-intersecting problem
Inequalities for matrices and the L-intersecting problem Richard M. Wilson California Institute of Technology Pasadena, CA 91125, USA Systems of Lines WPI, August 10, 2015 The L-intersecting problem Given
More informationFiedler s Theorems on Nodal Domains
Spectral Graph Theory Lecture 7 Fiedler s Theorems on Nodal Domains Daniel A Spielman September 9, 202 7 About these notes These notes are not necessarily an accurate representation of what happened in
More informationConvexity of the Joint Numerical Range
Convexity of the Joint Numerical Range Chi-Kwong Li and Yiu-Tung Poon October 26, 2004 Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his retirement. Abstract Let A = (A 1,..., A m ) be an
More informationLinear algebra and applications to graphs Part 1
Linear algebra and applications to graphs Part 1 Written up by Mikhail Belkin and Moon Duchin Instructor: Laszlo Babai June 17, 2001 1 Basic Linear Algebra Exercise 1.1 Let V and W be linear subspaces
More informationCSC Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming
CSC2411 - Linear Programming and Combinatorial Optimization Lecture 10: Semidefinite Programming Notes taken by Mike Jamieson March 28, 2005 Summary: In this lecture, we introduce semidefinite programming
More informationParity Versions of 2-Connectedness
Parity Versions of 2-Connectedness C. Little Institute of Fundamental Sciences Massey University Palmerston North, New Zealand c.little@massey.ac.nz A. Vince Department of Mathematics University of Florida
More informationStrictly Positive Definite Functions on the Circle
the Circle Princeton University Missouri State REU 2012 Definition A continuous function f : [0, π] R is said to be positive definite on S 1 if, for every N N and every set of N points x 1,..., x N on
More informationThe Matrix-Tree Theorem
The Matrix-Tree Theorem Christopher Eur March 22, 2015 Abstract: We give a brief introduction to graph theory in light of linear algebra. Our results culminates in the proof of Matrix-Tree Theorem. 1 Preliminaries
More informationA 2 G 2 A 1 A 1. (3) A double edge pointing from α i to α j if α i, α j are not perpendicular and α i 2 = 2 α j 2
46 MATH 223A NOTES 2011 LIE ALGEBRAS 11. Classification of semisimple Lie algebras I will explain how the Cartan matrix and Dynkin diagrams describe root systems. Then I will go through the classification
More informationThe Terwilliger algebra of a Q-polynomial distance-regular graph with respect to a set of vertices
The Terwilliger algebra of a Q-polynomial distance-regular graph with respect to a set of vertices Hajime Tanaka (joint work with Rie Tanaka and Yuta Watanabe) Research Center for Pure and Applied Mathematics
More informationDistance-regular graphs where the distance-d graph has fewer distinct eigenvalues
NOTICE: this is the author s version of a work that was accepted for publication in . Changes resulting from the publishing process, such as peer review, editing, corrections,
More informationThe complexity of counting graph homomorphisms
The complexity of counting graph homomorphisms Martin Dyer and Catherine Greenhill Abstract The problem of counting homomorphisms from a general graph G to a fixed graph H is a natural generalisation of
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationThis section is an introduction to the basic themes of the course.
Chapter 1 Matrices and Graphs 1.1 The Adjacency Matrix This section is an introduction to the basic themes of the course. Definition 1.1.1. A simple undirected graph G = (V, E) consists of a non-empty
More informationLecture Semidefinite Programming and Graph Partitioning
Approximation Algorithms and Hardness of Approximation April 16, 013 Lecture 14 Lecturer: Alantha Newman Scribes: Marwa El Halabi 1 Semidefinite Programming and Graph Partitioning In previous lectures,
More informationMath 775 Homework 1. Austin Mohr. February 9, 2011
Math 775 Homework 1 Austin Mohr February 9, 2011 Problem 1 Suppose sets S 1, S 2,..., S n contain, respectively, 2, 3,..., n 1 elements. Proposition 1. The number of SDR s is at least 2 n, and this bound
More informationPacking triangles in regular tournaments
Packing triangles in regular tournaments Raphael Yuster Abstract We prove that a regular tournament with n vertices has more than n2 11.5 (1 o(1)) pairwise arc-disjoint directed triangles. On the other
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 11 Luca Trevisan February 29, 2016
U.C. Berkeley CS294: Spectral Methods and Expanders Handout Luca Trevisan February 29, 206 Lecture : ARV In which we introduce semi-definite programming and a semi-definite programming relaxation of sparsest
More informationCHAPTER 2 -idempotent matrices
CHAPTER 2 -idempotent matrices A -idempotent matrix is defined and some of its basic characterizations are derived (see [33]) in this chapter. It is shown that if is a -idempotent matrix then it is quadripotent
More informationAverage mixing of continuous quantum walks
Average mixing of continuous quantum walks Juergen Kritschgau Department of Mathematics Iowa State University Ames, IA 50011 jkritsch@iastate.edu April 7, 2017 Juergen Kritschgau (ISU) 2 of 17 This is
More informationStatistical Mechanics and Combinatorics : Lecture IV
Statistical Mechanics and Combinatorics : Lecture IV Dimer Model Local Statistics We ve already discussed the partition function Z for dimer coverings in a graph G which can be computed by Kasteleyn matrix
More informationApplications of the Lopsided Lovász Local Lemma Regarding Hypergraphs
Regarding Hypergraphs Ph.D. Dissertation Defense April 15, 2013 Overview The Local Lemmata 2-Coloring Hypergraphs with the Original Local Lemma Counting Derangements with the Lopsided Local Lemma Lopsided
More informationRow and Column Distributions of Letter Matrices
College of William and Mary W&M ScholarWorks Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2016 Row and Column Distributions of Letter Matrices Xiaonan Hu College of William and
More informationVertex subsets with minimal width and dual width
Vertex subsets with minimal width and dual width in Q-polynomial distance-regular graphs University of Wisconsin & Tohoku University February 2, 2011 Every face (or facet) of a hypercube is a hypercube...
More informationLEVEL MATRICES G. SEELINGER, P. SISSOKHO, L. SPENCE, AND C. VANDEN EYNDEN
LEVEL MATRICES G. SEELINGER, P. SISSOKHO, L. SPENCE, AND C. VANDEN EYNDEN Abstract. Let n > 1 and k > 0 be fixed integers. A matrix is said to be level if all its column sums are equal. A level matrix
More informationON COST MATRICES WITH TWO AND THREE DISTINCT VALUES OF HAMILTONIAN PATHS AND CYCLES
ON COST MATRICES WITH TWO AND THREE DISTINCT VALUES OF HAMILTONIAN PATHS AND CYCLES SANTOSH N. KABADI AND ABRAHAM P. PUNNEN Abstract. Polynomially testable characterization of cost matrices associated
More informationCombinatorial Optimization
Combinatorial Optimization Problem set 8: solutions 1. Fix constants a R and b > 1. For n N, let f(n) = n a and g(n) = b n. Prove that f(n) = o ( g(n) ). Solution. First we observe that g(n) 0 for all
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 informationIntroduction to Semidefinite Programming I: Basic properties a
Introduction to Semidefinite Programming I: Basic properties and variations on the Goemans-Williamson approximation algorithm for max-cut MFO seminar on Semidefinite Programming May 30, 2010 Semidefinite
More informationLaplacian Integral Graphs with Maximum Degree 3
Laplacian Integral Graphs with Maximum Degree Steve Kirkland Department of Mathematics and Statistics University of Regina Regina, Saskatchewan, Canada S4S 0A kirkland@math.uregina.ca Submitted: Nov 5,
More informationSince G is a compact Lie group, we can apply Schur orthogonality to see that G χ π (g) 2 dg =
Problem 1 Show that if π is an irreducible representation of a compact lie group G then π is also irreducible. Give an example of a G and π such that π = π, and another for which π π. Is this true for
More informationRemoval Lemmas with Polynomial Bounds
Removal Lemmas with Polynomial Bounds Lior Gishboliner Asaf Shapira Abstract A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. One
More information3.3 Easy ILP problems and totally unimodular matrices
3.3 Easy ILP problems and totally unimodular matrices Consider a generic ILP problem expressed in standard form where A Z m n with n m, and b Z m. min{c t x : Ax = b, x Z n +} (1) P(b) = {x R n : Ax =
More informationMaximizing the numerical radii of matrices by permuting their entries
Maximizing the numerical radii of matrices by permuting their entries Wai-Shun Cheung and Chi-Kwong Li Dedicated to Professor Pei Yuan Wu. Abstract Let A be an n n complex matrix such that every row and
More information1 Strict local optimality in unconstrained optimization
ORF 53 Lecture 14 Spring 016, Princeton University Instructor: A.A. Ahmadi Scribe: G. Hall Thursday, April 14, 016 When in doubt on the accuracy of these notes, please cross check with the instructor s
More informationRobust Principal Component Analysis
ELE 538B: Mathematics of High-Dimensional Data Robust Principal Component Analysis Yuxin Chen Princeton University, Fall 2018 Disentangling sparse and low-rank matrices Suppose we are given a matrix M
More informationLimitations in Approximating RIP
Alok Puranik Mentor: Adrian Vladu Fifth Annual PRIMES Conference, 2015 Outline 1 Background The Problem Motivation Construction Certification 2 Planted model Planting eigenvalues Analysis Distinguishing
More information= ϕ r cos θ. 0 cos ξ sin ξ and sin ξ cos ξ. sin ξ 0 cos ξ
8. The Banach-Tarski paradox May, 2012 The Banach-Tarski paradox is that a unit ball in Euclidean -space can be decomposed into finitely many parts which can then be reassembled to form two unit balls
More informationDominating a family of graphs with small connected subgraphs
Dominating a family of graphs with small connected subgraphs Yair Caro Raphael Yuster Abstract Let F = {G 1,..., G t } be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive
More informationHadamard matrices and strongly regular graphs with the 3-e.c. adjacency property
Hadamard matrices and strongly regular graphs with the 3-e.c. adjacency property Anthony Bonato Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada, N2L 3C5 abonato@wlu.ca
More informationThe Algorithmic Aspects of the Regularity Lemma
The Algorithmic Aspects of the Regularity Lemma N. Alon R. A. Duke H. Lefmann V. Rödl R. Yuster Abstract The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in
More informationDISJOINT PAIRS OF SETS AND INCIDENCE MATRICES
DISJOINT PAIRS OF SETS AND INCIDENCE MATRICES B MARVIN MARCUS AND HENRK MINC In a recent paper [1] investigating the repeated appearance of zeros in the powers of a mtrix the following purely combinatorial
More informationTopics in Graph Theory
Topics in Graph Theory September 4, 2018 1 Preliminaries A graph is a system G = (V, E) consisting of a set V of vertices and a set E (disjoint from V ) of edges, together with an incidence function End
More informationOn the Non-Symmetric Spectra of Certain Graphs
College of William and Mary W&M Publish College of William & Mary Undergraduate Honors Theses Theses, Dissertations, & Master Projects 5-2015 On the Non-Symmetric Spectra of Certain Graphs Owen Hill College
More informationIntrinsic products and factorizations of matrices
Available online at www.sciencedirect.com Linear Algebra and its Applications 428 (2008) 5 3 www.elsevier.com/locate/laa Intrinsic products and factorizations of matrices Miroslav Fiedler Academy of Sciences
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationTropical Constructions and Lifts
Tropical Constructions and Lifts Hunter Ash August 27, 2014 1 The Algebraic Torus and M Let K denote a field of characteristic zero and K denote the associated multiplicative group. A character on (K )
More informationWhen can perfect state transfer occur?
Electronic Journal of Linear Algebra Volume 23 Volume 23 (2012) Article 62 2012 When can perfect state transfer occur? Chris Godsil cgodsil@uwaterloo.ca Follow this and additional works at: http://repository.uwyo.edu/ela
More informationMath/CS 466/666: Homework Solutions for Chapter 3
Math/CS 466/666: Homework Solutions for Chapter 3 31 Can all matrices A R n n be factored A LU? Why or why not? Consider the matrix A ] 0 1 1 0 Claim that this matrix can not be factored A LU For contradiction,
More informationTheorems of Erdős-Ko-Rado type in polar spaces
Theorems of Erdős-Ko-Rado type in polar spaces Valentina Pepe, Leo Storme, Frédéric Vanhove Department of Mathematics, Ghent University, Krijgslaan 28-S22, 9000 Ghent, Belgium Abstract We consider Erdős-Ko-Rado
More information9.1 Eigenvectors and Eigenvalues of a Linear Map
Chapter 9 Eigenvectors and Eigenvalues 9.1 Eigenvectors and Eigenvalues of a Linear Map Given a finite-dimensional vector space E, letf : E! E be any linear map. If, by luck, there is a basis (e 1,...,e
More informationCLASSIFICATION OF TREES EACH OF WHOSE ASSOCIATED ACYCLIC MATRICES WITH DISTINCT DIAGONAL ENTRIES HAS DISTINCT EIGENVALUES
Bull Korean Math Soc 45 (2008), No 1, pp 95 99 CLASSIFICATION OF TREES EACH OF WHOSE ASSOCIATED ACYCLIC MATRICES WITH DISTINCT DIAGONAL ENTRIES HAS DISTINCT EIGENVALUES In-Jae Kim and Bryan L Shader Reprinted
More informationAn Interlacing Approach for Bounding the Sum of Laplacian Eigenvalues of Graphs
An Interlacing Approach for Bounding the Sum of Laplacian Eigenvalues of Graphs Tilburg University joint work with M.A. Fiol, W.H. Haemers and G. Perarnau Laplacian matrix Eigenvalue interlacing Two cases
More informationa factors The exponential 0 is a special case. If b is any nonzero real number, then
0.1 Exponents The expression x a is an exponential expression with base x and exponent a. If the exponent a is a positive integer, then the expression is simply notation that counts how many times the
More informationKatarzyna Mieczkowska
Katarzyna Mieczkowska Uniwersytet A. Mickiewicza w Poznaniu Erdős conjecture on matchings in hypergraphs Praca semestralna nr 1 (semestr letni 010/11 Opiekun pracy: Tomasz Łuczak ERDŐS CONJECTURE ON MATCHINGS
More informationb jσ(j), Keywords: Decomposable numerical range, principal character AMS Subject Classification: 15A60
On the Hu-Hurley-Tam Conjecture Concerning The Generalized Numerical Range Che-Man Cheng Faculty of Science and Technology, University of Macau, Macau. E-mail: fstcmc@umac.mo and Chi-Kwong Li Department
More informationArithmetic Progressions with Constant Weight
Arithmetic Progressions with Constant Weight Raphael Yuster Department of Mathematics University of Haifa-ORANIM Tivon 36006, Israel e-mail: raphy@oranim.macam98.ac.il Abstract Let k n be two positive
More informationLecture 4 Orthonormal vectors and QR factorization
Orthonormal vectors and QR factorization 4 1 Lecture 4 Orthonormal vectors and QR factorization EE263 Autumn 2004 orthonormal vectors Gram-Schmidt procedure, QR factorization orthogonal decomposition induced
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 informationShortest paths with negative lengths
Chapter 8 Shortest paths with negative lengths In this chapter we give a linear-space, nearly linear-time algorithm that, given a directed planar graph G with real positive and negative lengths, but no
More informationEventually reducible matrix, eventually nonnegative matrix, eventually r-cyclic
December 15, 2012 EVENUAL PROPERIES OF MARICES LESLIE HOGBEN AND ULRICA WILSON Abstract. An eventual property of a matrix M C n n is a property that holds for all powers M k, k k 0, for some positive integer
More informationGraph fundamentals. Matrices associated with a graph
Graph fundamentals Matrices associated with a graph Drawing a picture of a graph is one way to represent it. Another type of representation is via a matrix. Let G be a graph with V (G) ={v 1,v,...,v n
More informationELA
SUBDOMINANT EIGENVALUES FOR STOCHASTIC MATRICES WITH GIVEN COLUMN SUMS STEVE KIRKLAND Abstract For any stochastic matrix A of order n, denote its eigenvalues as λ 1 (A),,λ n(a), ordered so that 1 = λ 1
More informationQuasi-randomness of graph balanced cut properties
Quasi-randomness of graph balanced cut properties Hao Huang Choongbum Lee Abstract Quasi-random graphs can be informally described as graphs whose edge distribution closely resembles that of a truly random
More informationMatching Polynomials of Graphs
Spectral Graph Theory Lecture 25 Matching Polynomials of Graphs Daniel A Spielman December 7, 2015 Disclaimer These notes are not necessarily an accurate representation of what happened in class The notes
More informationAn Introduction to Spectral Graph Theory
An Introduction to Spectral Graph Theory Mackenzie Wheeler Supervisor: Dr. Gary MacGillivray University of Victoria WheelerM@uvic.ca Outline Outline 1. How many walks are there from vertices v i to v j
More informationZ-Pencils. November 20, Abstract
Z-Pencils J. J. McDonald D. D. Olesky H. Schneider M. J. Tsatsomeros P. van den Driessche November 20, 2006 Abstract The matrix pencil (A, B) = {tb A t C} is considered under the assumptions that A is
More informationAALBORG UNIVERSITY. Directed strongly regular graphs with rank 5. Leif Kjær Jørgensen. Department of Mathematical Sciences. Aalborg University
AALBORG UNIVERSITY Directed strongly regular graphs with ran 5 by Leif Kjær Jørgensen R-2014-05 May 2014 Department of Mathematical Sciences Aalborg University Fredri Bajers Vej G DK - 9220 Aalborg Øst
More informationIntroduction to Kleene Algebras
Introduction to Kleene Algebras Riccardo Pucella Basic Notions Seminar December 1, 2005 Introduction to Kleene Algebras p.1 Idempotent Semirings An idempotent semiring is a structure S = (S, +,, 1, 0)
More informationMinimum number of non-zero-entries in a 7 7 stable matrix
Linear Algebra and its Applications 572 (2019) 135 152 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Minimum number of non-zero-entries in a
More informationApplying block intersection polynomials to study graphs and designs
Applying block intersection polynomials to study graphs and designs Leonard Soicher Queen Mary University of London CoCoA15, Colorado State University, Fort Collins, July 2015 Leonard Soicher (QMUL) Block
More informationStatistical Inference on Large Contingency Tables: Convergence, Testability, Stability. COMPSTAT 2010 Paris, August 23, 2010
Statistical Inference on Large Contingency Tables: Convergence, Testability, Stability Marianna Bolla Institute of Mathematics Budapest University of Technology and Economics marib@math.bme.hu COMPSTAT
More informationThe Turán number of sparse spanning graphs
The Turán number of sparse spanning graphs Noga Alon Raphael Yuster Abstract For a graph H, the extremal number ex(n, H) is the maximum number of edges in a graph of order n not containing a subgraph isomorphic
More informationThe third smallest eigenvalue of the Laplacian matrix
Electronic Journal of Linear Algebra Volume 8 ELA Volume 8 (001) Article 11 001 The third smallest eigenvalue of the Laplacian matrix Sukanta Pati pati@iitg.ernet.in Follow this and additional works at:
More informationMinimum rank of a graph over an arbitrary field
Electronic Journal of Linear Algebra Volume 16 Article 16 2007 Minimum rank of a graph over an arbitrary field Nathan L. Chenette Sean V. Droms Leslie Hogben hogben@aimath.org Rana Mikkelson Olga Pryporova
More information