# Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002

Save this PDF as:

Size: px
Start display at page:

Download "Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002"

## Transcription

1 Lecture notes from Foundations of Markov chain Monte Carlo methods University of Chicago, Spring 2002 Lecture 1, March 29, 2002 Eric Vigoda Scribe: Varsha Dani & Tom Hayes 1.1 Introduction The aim of this course is to address the complexity of counting and sampling problems from an algorithmic perspective. Typically we will be interested in counting the size of a collection of combinatorial structures of a graph, e.g., the number of spanning trees of a graph. We will soon see that this style of counting problem is intimately related to the sampling problem which asks for a random structure from this collection, e.g., generate a random spanning tree. The course will demonstrate that the class of counting and sampling problems can be addressed in a very cohesive and elegant (at least to my tastes) manner. In this lecture we study some classical algorithms for exact counting. There are few problems which admit efficient algorithms for the exact counting version. In most cases we will have to settle for approximation algorithms. It is interesting to note that both of the algorithms presented in this lecture rely on a reduction to the determinant. This is the case for virtually all (of the very few known) exact counting algorithms. For the two problems considered in this lecture we will show a reduction to the determinant. Consequently both of these problems can be solved in O(n 3 ) time where n is the number of vertices in the input graph. 1.2 Spanning Trees Our first theorem is known as Kirchoff s Matrix-Tree Theorem [2], and dates back over 150 years. We are interested in counting the number of spanning trees of an arbitrary undirected graph G = (V, E) with no self-loops. Assume the graph is given by its adjacency matrix A where A(i, j) = { 1 if (i, j) E 0 otherwise Associate the vertex set V with [n] = {0,..., n 1}, and let D be the diagonal matrix with entry D(i, i) equal to the degree of vertex i for all i [n]. It will be useful to consider a corresponding problem on directed graphs. 1-1

2 1-2 Definition 1.1 For a directed graph G = (V, E ), a set T E is an out-tree rooted at r V if the subgraph (V, T ) has the following properties: it is acyclic (no directed cycles), the in-degree of r is 0, and the in-degree of every vertex except r is exactly 1. Taken together these conditions imply that, given an out-tree T, ignoring orientations the resulting undirected graph is still acyclic. We leave the proof of this observation to the reader. We are interested in the natural directed graph version of our undirected graph. Let G = (V, E ) denote the directed graph formed by replacing each edge (i, j) E by a pair of anti-symmetric directed edges ij and ji. The following claim illustrates that we can reduce the problem of counting spanning trees to counting out-trees. Claim 1.2 For any r V, the number of spanning trees of G equals the number of out-trees rooted at r in G. Proof: We prove the claim by showing a bijection between the collections. Fix a root r V. From an out-tree T with root r, form the associated spanning tree by simply removing the orientations on the edges. As we claimed earlier, the resulting undirected graph is acyclic and thus a snapping tree. This mapping can be reversed, hence it is a bijection. The reverse mapping is the following: Given a spanning tree simply orient each edge away from r (there is a unique path from r to any edge). We need some additional notation before stating the main theorem. Recall, A denotes the adjacency matrix of the undirected graph G = (V, E), and n = V. Let K = D A denote the Laplacian matrix of G. Specifically, K is an n n matrix where deg(i) if i = j K(i, j) = 1 if (i, j) E 0 otherwise The matrix K is known as the Kirchoff in-degree matrix. Let K i,i be the minor of K obtained by deleting row i and column i. Theorem 1.3 ([2]) For any G = (V, E) and r V, the number of out-trees rooted at r in G equals det(k r,r ). Proof: Without loss of generality, let r = n 1. Recalling the definition of the determinant, we have det(k n 1,n 1 ) = sgn(σ) K(i, σ(i)), σ S n 1 0 i n 2

3 1-3 where sgn(σ) = ( 1) #(even cycles in σ) and S n 1 is the set of permutations of [n 1] = {0, 1,..., n 2}. It will be useful to consider another equivalent form of the sign of the permutation. For a cycle C in σ we have sgn(c) = ( 1) C +1 so that it is 1 for even-length cycles, and +1 for odd-length cycles. Hence, multiplying through all cycles in σ, we have sgn(σ) = ( 1) σ +#(cycles in σ) = ( 1) n 1+#(cycles in σ) We proceed by decomposing σ based on its fixed points, i.e., self-loops, let L = L(σ) = {i : σ(i) = i}, and L = [n 1] \ L. We first decompose the inner product: K(i, σ(i)) = K(i, i) K(i, σ(i)) 0 i n 2 = deg(i) A(i, σ(i)) = = 0 j n 1 0 j n 1 A(i, j) ( 1) L A(i, σ(i)) A(j, i) ( 1) L A(i, σ(i)). The last line uses the fact that G is undirected, hence A is symmetric. Now we ll add in the sign of the permutation, and use the fact that the number of cycles in σ equals the cardinality of L plus the number of cycles in L: sgn(σ) K(i, σ(i)) = ( 1) n 1+#(cycles in σ) A(j, i) ( 1) L A(i, σ(i)) i 0 j n 1 = ( 1) n 1+ L + L +#(cycles in L) A(j, i) A(i, σ(i)) 1 j n = ( 1) 2(n 1)+#(cycles in L) A(j, i) A(i, σ(i)) 1 j n = ( 1) #(cycles in L) A(j, i) A(i, σ(i)). 1 j n

4 1-4 Finally, we would like to change the order of the summation and product over the vertices in L. We have: sgn(σ) K(i, σ(i)) = ( 1) #(cycles in L) A(f(i), i) A(i, σ(i)) i f:l [n] For i L, we can consider (i, σ(i)) as a possible directed edge. We only need to consider permutations σ where all of these edges exist, otherwise the term is 0. These edges define a directed cycle cover of the vertices L. Similarly, we can consider the directed graph defined by the edges f(i), i for all i L. This is a subgraph where each vertex i L is assigned one parent. We can rewrite the above sum over permutations as a sum over the appropriate subgraphs. The corresponding subgraphs are S E where every vertex i L has in-degree 1, and S E which is a cycle cover of L. For T E, let w(t ) = (i,j) T A(i, j). Then we have det(k n 1,n 1 ) = ( 1) #(cycles in S ) w(s)w(s ). (L,S,S ) When L = n 1 and S is acyclic, then S is an out-tree of G rooted at r. We can partition the previous sum with the aim of eliminating any terms in which L n 1 and/or S contains a cycle. Let c and c denote the number of cycles in S and S, respectively, and let C min denote the cycle in S S which contains the smallest numbered vertex. det(k n 1,n 1 ) = = (L, S, S ) : L = n 1, c = 0 w(s) + (L, S, S ) : L = n 1, c = 0 w(s) + ( 1) c w(s)w(s ) (L, S L, S L ) : c + c > 0 ( 1) c w(s)w(s ) + (L, S, S ) : c + c > 0, C min S (L, S, S ) : c + c > 0, C min S ( 1) c w(s)w(s ) There is a sign-reversing, magnitude-preserving bijection between the terms of the last two summations. In particular, move the cycle C min between S and S, simultaneously moving all of the vertices of C min between L and L. This bijection implies these two sums cancel each other out, and we are left with the sum over possible out-trees. Since the term w(s) is 1 if all the edges exist and 0 otherwise, we have proven det(k n 1,n 1 ) equals the number of out-trees rooted at r. 1.3 Permanent of planar graphs The permanent is a natural variant of the determinant without the alternating sign. The formal definition is as follows.

5 1-5 Definition 1.4 For a n n matrix A, let per(a) = a i,σ(i). σ S n i If A is a 0/1 matrix, then the permanent equals the number of perfect matchings in the bipartite graph G with incidence matrix A. To be precise, the bipartite graph has 2n vertices where each vertex is associated with a specific row or column of A. The vertex for row i and the vertex for column j if a i,j = 1. A permutation corresponds to a pairing of row and column vertices. If all of the edges in this pairing exist, and therefore defines a perfect matching, then the permutation contributes 1 to per(a), and contributes 0 if any edge in the pairing does not exist in G. Our goal is to compute the permanent for a large class of matrices. For simplicity, we will focus on 0/1 matrices, however everything generalizes to arbitrary non-negative weights on the edges. We shift attention to the graph theoretic problem of perfect matchings and consider the class of planar (not necessarily bipartite) graphs. We present Kasteleyn s algorithm for computing the number of perfect matchings of a planar graph G. The main idea is to orient the edges of G, resulting in some G, so that the number of perfect matchings equals the square root of the determinant of the adjacency matrix of G. Before embarking, we begin with a basic, critical observation about perfect matchings. Claim 1.5 For a pair of perfect matchings M, M, the collection M M consists of disjoint even length cycles and edges. An orientation G of G is an assignment of a unique direction to each edge. We are aiming for a Pfaffian orientation, which we now formally define. Definition 1.6 An even length cycle is oddly oriented in G, if there are an odd # of clockwise edges. Definition 1.7 An orientation G of an undirected graph G is Pfaffian if for all perfect matchings M, M, all cycles of M M are oddly oriented. The previous definition is well-defined since all cycles in M M are even length. The main result considers the adjacency matrix of an orientation G. Definition 1.8 For an orientation G= (V, E), let A = (a ij ) where 1 if ij E a i,j = 1 if ji E 0 otherwise

6 1-6 The following theorem is a major component of Kasteleyn s algorithm. Theorem 1.9 ([1]) For any Pfaffian orientation G of G, the square of the number of perfect matchings of G equals the determinant of A( G). Before proving the theorem, we convert the problem to an associated problem on directed graphs. Recall the definition of G = (V, E ) from our earlier work on counting spanning trees. In the following lemma, an even cycle cover is a set of (vertex) disjoint directed even length cycles. Lemma 1.10 The square of the number of perfect matchings of G equals the number of even cycle covers of G. Proof: We define a bijection between ordered pairs of perfect matchings and even cycle covers. Consider an ordered pair of perfect matchings M, M and we will define their associated cycle cover. Each cycle C in M M is oriented in a canonical way in order to insure the mapping is invertible. Specifically, take the smallest vertex v in C and orient the edge of M incident to v away from v. This forces a unique orientation for the remaining edges of C. Observe that this mapping can easily be reversed, and the lemma is proven. Proof of Theorem: Recall the definition of the determinant, det(a( G)) = sgn(σ) σ S n i a i,σ(i) Decompose each permutation into its constituent cycles σ = γ 1... γ k, and let V i V denote the vertices in cycle γ i. We claim the set of permutations which contain some odd-length cycle cancel each other out. To prove this claim, consider a permutation σ and its first odd-length cycle γ i. Let (we have simply reversed the i-th cycle). σ = γ 1 γ i 1 γi 1 γ i+1 γ k Since σ and σ contain the same number of cycles, sgn(σ) = sgn(σ ). Since γ i is odd-length, observe that the number of clockwise edges is of opposite parity to the number of counter-clockwise edges, hence a j,γi (j) = a j,γ 1 i (j). j V i j V i Therefore, the contributions from σ and σ sum to 0. Consider a permutation σ where every cycle γ i is even length. We have j V i a j,σ(j) = 1, for all i, since the orientation is Pfaffian. Thus, a l,σ(l) = ( 1) k = sgn(σ), l V

7 1-7 where k is the number of cycles in σ (all of which are even-length). Therefore, each permutation consisting of only even length cycles contributes 1 to the summation. Since every such permutation corresponds to an even cycle cover of G, we have proven det(a( G)) equals the number of even cycle covers of G. This completes the proof of the theorem. We still need to find a Pfaffian orientation for planar graphs. The following structural lemma is the essential ingredient. Lemma 1.11 For a planar graph G with orientation G, (i) if every face (except possibly the outer face) has an odd number of clockwise edges, then (ii) for all cycles C the number of clockwise edges of C is of opposite parity to the number of vertices in the interior of C, which implies (iii) G is a Pfaffian orientation. Proof: We begin by proving (ii) = (iii). Consider a cycle C contained in M M for some pair of perfect matchings M, M. If any vertex in the interior of C is matched in M with a vertex outside C, then planarity is contradicted. Therefore, there must be an even number of vertices inside C, and (iii) then follows. We now prove (i) implies (ii). Consider a cycle C of G, and the induced subgraph of the vertices on and inside C. Let F IN denote the number of non-outer faces in this subgraph, and let f 1,..., f FIN denote the associated faces. Let V ON (respectively, E ON ) denote the number of vertices (edges) on C. Similarly define V IN and E IN for the vertices and edges strictly inside C. Finally, let EON CLOCK (f i ) (respectively, EON CLOCK (C)) denote the number of clockwise edges on face f i (cycle C). Plugging the terms into Euler s formula we have Since C is a cycle, V ON = E ON. This implies, (V ON + V IN ) (E ON + E IN ) + (F IN + 1) = 2. V IN E IN + F IN = 1. (1.1) By (i), for every face f i, E CLOCK ON (f i ) 1 (mod 2). Summing over all faces inside C, F IN i E CLOCK ON (f i ) (mod 2). (1.2) Since each edge inside the cycle is clockwise to exactly one of the two faces incident it and each clockwise edge on C is clockwise to the inside face incident it, we have E IN + EON CLOCK (C) = i E CLOCK ON (f i ) (1.3)

8 1-8 Combining (1.2) with (1.3) yields Applying (1.1) we have or equivalently This proves the lemma. F IN E CLOCK ON (C) + E IN (mod 2). F IN E CLOCK ON (C) + F IN + V IN 1, E CLOCK ON (C) + V IN 1 (mod 2). We are now in position to easily find an Pfaffian orientation of a planar graph. The algorithm constructs the orientation inductively on the number of edges (conditioned on the graph being connected). The base case is a tree. For a tree, the only face is the outer face, hence every edge can be oriented arbitrarily. For a general planar graph, consider an edge e lying on the outer face. Inductively orient G \ e. Adding e creates one new face f. One of the two orientations of e must guarantee the correct parity for the number of clockwise edges on f. This completes the algorithm for constructing a Pfaffian orientation. Remark 1.12 For bipartite graphs, McCuaig, Robertson, Seymour and Thomas [3] devised an algorithm which determines if the input graph has a Pfaffian orientation (and constructing such an orientation if it exists). The problem for general graphs is still open. References [1] P. W. Kasteleyn. Graph theory and crystal physics. In Graph Theory and Theoretical Physics, pages Academic Press, London, [2] G. Kirchoff. Über die ausflösung der gleichungen auf welche man bei der untersuchungen der linearen verteilung galvanisher ströme geführt wird. Poggendorf Ann. Physik, 72: , [3] William McCuaig, Neil Robertson, Paul D. Seymour, and Robin Thomas. Permanents, Pfaffian orientations, and even directed circuits (extended abstract). In STOC, pages , 1997.

### Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the

### Homework 15 Solutions

PROBLEM ONE (Trees) Homework 15 Solutions 1. Recall the definition of a tree: a tree is a connected, undirected graph which has no cycles. Which of the following definitions are equivalent to this definition

### Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

### Social Media Mining. Graph Essentials

Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

### Midterm Practice Problems

6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

### Graph Classification and Easy Reliability Polynomials

Mathematical Assoc. of America American Mathematical Monthly 121:1 November 18, 2014 1:11 a.m. AMM.tex page 1 Graph Classification and Easy Reliability Polynomials Pablo Romero and Gerardo Rubino Abstract.

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### Lecture 7. The Matrix-Tree Theorem. 7.1 Kircho s Matrix-Tree Theorem

Lecture The Matrix-Tree Theorem This section of the notes introduces a very beautiful theorem that uses linear algebra to count trees in graphs. Reading: The next few lectures are not covered in Jungnickel

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

### 3. Markov property. Markov property for MRFs. Hammersley-Clifford theorem. Markov property for Bayesian networks. I-map, P-map, and chordal graphs

Markov property 3-3. Markov property Markov property for MRFs Hammersley-Clifford theorem Markov property for ayesian networks I-map, P-map, and chordal graphs Markov Chain X Y Z X = Z Y µ(x, Y, Z) = f(x,

### Chapter 4: Binary Operations and Relations

c Dr Oksana Shatalov, Fall 2014 1 Chapter 4: Binary Operations and Relations 4.1: Binary Operations DEFINITION 1. A binary operation on a nonempty set A is a function from A A to A. Addition, subtraction,

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### 4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests

4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

### 1 Digraphs. Definition 1

1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### Planar Graph and Trees

Dr. Nahid Sultana December 16, 2012 Tree Spanning Trees Minimum Spanning Trees Maps and Regions Eulers Formula Nonplanar graph Dual Maps and the Four Color Theorem Tree Spanning Trees Minimum Spanning

### CSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24

CSL851: Algorithmic Graph Theory Semester I 2013-2014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have

### Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010

Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

CS-62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of

### Min-cost flow problems and network simplex algorithm

Min-cost flow problems and network simplex algorithm The particular structure of some LP problems can be sometimes used for the design of solution techniques more efficient than the simplex algorithm.

### Spectral graph theory

Spectral graph theory Uri Feige January 2010 1 Background With every graph (or digraph) one can associate several different matrices. We have already seen the vertex-edge incidence matrix, the Laplacian

### ORDERS OF ELEMENTS IN A GROUP

ORDERS OF ELEMENTS IN A GROUP KEITH CONRAD 1. Introduction Let G be a group and g G. We say g has finite order if g n = e for some positive integer n. For example, 1 and i have finite order in C, since

### A counterexample to a result on the tree graph of a graph

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 63(3) (2015), Pages 368 373 A counterexample to a result on the tree graph of a graph Ana Paulina Figueroa Departamento de Matemáticas Instituto Tecnológico

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### Discrete Mathematics Problems

Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................

### Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest.

2 Trees What is a tree? Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest. Theorem 2.1 If G is a forest, then comp(g) = V (G) E(G). Proof: We proceed by induction on E(G).

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### Relations Graphical View

Relations Slides by Christopher M. Bourke Instructor: Berthe Y. Choueiry Introduction Recall that a relation between elements of two sets is a subset of their Cartesian product (of ordered pairs). A binary

### Planarity Planarity

Planarity 8.1 71 Planarity Up until now, graphs have been completely abstract. In Topological Graph Theory, it matters how the graphs are drawn. Do the edges cross? Are there knots in the graph structure?

### 6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

### *.I Zolotareff s Proof of Quadratic Reciprocity

*.I. ZOLOTAREFF S PROOF OF QUADRATIC RECIPROCITY 1 *.I Zolotareff s Proof of Quadratic Reciprocity This proof requires a fair amount of preparations on permutations and their signs. Most of the material

### Electrical Resistances in Products of Graphs

Electrical Resistances in Products of Graphs By Shelley Welke Under the direction of Dr. John S. Caughman In partial fulfillment of the requirements for the degree of: Masters of Science in Teaching Mathematics

### Chapter 4: Trees. 2. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

9 Properties of Trees. Definitions: Chapter 4: Trees forest - a graph that contains no cycles tree - a connected forest. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

### The tree-number and determinant expansions (Biggs 6-7)

The tree-number and determinant expansions (Biggs 6-7) André Schumacher March 20, 2006 Overview Biggs 6-7 [1] The tree-number κ(γ) κ(γ) and the Laplacian matrix The σ function Elementary (sub)graphs Coefficients

### Week 5: Binary Relations

1 Binary Relations Week 5: Binary Relations The concept of relation is common in daily life and seems intuitively clear. For instance, let X be the set of all living human females and Y the set of all

### Chapter 7. Permutation Groups

Chapter 7 Permutation Groups () We started the study of groups by considering planar isometries In the previous chapter, we learnt that finite groups of planar isometries can only be cyclic or dihedral

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### A simple proof for the number of tilings of quartered Aztec diamonds

A simple proof for the number of tilings of quartered Aztec diamonds arxiv:1309.6720v1 [math.co] 26 Sep 2013 Tri Lai Department of Mathematics Indiana University Bloomington, IN 47405 tmlai@indiana.edu

### Long questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and -3B.

Unit-1: Matrix Algebra Short questions answer 1. What is Matrix? 2. Define the following terms : a) Elements matrix b) Row matrix c) Column matrix d) Diagonal matrix e) Scalar matrix f) Unit matrix OR

### Coloring Eulerian triangulations of the projective plane

Coloring Eulerian triangulations of the projective plane Bojan Mohar 1 Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia bojan.mohar@uni-lj.si Abstract A simple characterization

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### Determinants LECTURE Calculating the Area of a Parallelogram. Definition Let A be a 2 2 matrix. A = The determinant of A is the number

LECTURE 13 Determinants 1. Calculating the Area of a Parallelogram Definition 13.1. Let A be a matrix. [ a c b d ] The determinant of A is the number det A) = ad bc Now consider the parallelogram formed

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH

DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,

### CMSC 451: Graph Properties, DFS, BFS, etc.

CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia

GLASNIK MATEMATIČKI Vol. 38(58)(2003), 2 232 MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia

### Some Results on 2-Lifts of Graphs

Some Results on -Lifts of Graphs Carsten Peterson Advised by: Anup Rao August 8, 014 1 Ramanujan Graphs Let G be a d-regular graph. Every d-regular graph has d as an eigenvalue (with multiplicity equal

### 2.1: Determinants by Cofactor Expansion. Math 214 Chapter 2 Notes and Homework. Evaluate a Determinant by Expanding by Cofactors

2.1: Determinants by Cofactor Expansion Math 214 Chapter 2 Notes and Homework Determinants The minor M ij of the entry a ij is the determinant of the submatrix obtained from deleting the i th row and the

### Complexity and Completeness of Finding Another Solution and Its Application to Puzzles

yato@is.s.u-tokyo.ac.jp seta@is.s.u-tokyo.ac.jp Π (ASP) Π x s x s ASP Ueda Nagao n n-asp parsimonious ASP ASP NP Complexity and Completeness of Finding Another Solution and Its Application to Puzzles Takayuki

### Combinatorics: The Fine Art of Counting

Combinatorics: The Fine Art of Counting Week 9 Lecture Notes Graph Theory For completeness I have included the definitions from last week s lecture which we will be using in today s lecture along with

### A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

### DO NOT RE-DISTRIBUTE THIS SOLUTION FILE

Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs

Graph Isomorphism Completeness for Perfect Graphs and Subclasses of Perfect Graphs C. Boucher D. Loker May 2006 Abstract A problem is said to be GI-complete if it is provably as hard as graph isomorphism;

### princeton university F 02 cos 597D: a theorist s toolkit Lecture 7: Markov Chains and Random Walks

princeton university F 02 cos 597D: a theorist s toolkit Lecture 7: Markov Chains and Random Walks Lecturer: Sanjeev Arora Scribe:Elena Nabieva 1 Basics A Markov chain is a discrete-time stochastic process

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Counting spanning trees

Counting spanning trees Question Given a graph G, howmanyspanningtreesdoesg have? (G) =numberofdistinctspanningtreesofg Definition If G =(V,E) isamultigraphwithe 2 E, theng e (said G contract e ) is the

### A CHARACTERIZATION OF TREE TYPE

A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying

### Perfect Matchings and Pfaffian Orientation

Perfect Matchings and Pfaffian Orientation Jeanette Nguyen Bachelor Thesis Supervisor: dr. Dion Gijswijt KdV Instituut voor wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### Section 6.4 Closures of Relations

Section 6.4 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 ordered pairs to R to obtain property P. In

### Analyzing the Smarts Pyramid Puzzle

Analyzing the Smarts Pyramid Puzzle Spring 2016 Analyzing the Smarts Pyramid Puzzle 1/ 22 Outline 1 Defining the problem and its solutions 2 Introduce and prove Burnside s Lemma 3 Proving the results Analyzing

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)

Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph

### Exponential time algorithms for graph coloring

Exponential time algorithms for graph coloring Uriel Feige Lecture notes, March 14, 2011 1 Introduction Let [n] denote the set {1,..., k}. A k-labeling of vertices of a graph G(V, E) is a function V [k].

### CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

### Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses

University of Warwick, EC9A0 Maths for Economists Peter J. Hammond 1 of 57 Lecture Notes 1: Matrix Algebra Part B: Determinants and Inverses Peter J. Hammond email: p.j.hammond@warwick.ac.uk Autumn 2012,

### An important class of codes are linear codes in the vector space Fq n, where F q is a finite field of order q.

Chapter 3 Linear Codes An important class of codes are linear codes in the vector space Fq n, where F q is a finite field of order q. Definition 3.1 (Linear code). A linear code C is a code in Fq n for

### Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible:

Cramer s Rule and the Adjugate Using determinants, it is possible to express the solution to a system of equations whose coefficient matrix is invertible: Theorem [Cramer s Rule] If A is an invertible

### Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania

Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics

### Graphs and Network Flows IE411 Lecture 1

Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture

### Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing:

1. GRAPHS AND COLORINGS Definition. A graph is a collection of vertices, and edges between them. They are often represented by a drawing: 3 vertices 3 edges 4 vertices 4 edges 4 vertices 6 edges A graph

### On minimal forbidden subgraphs for the class of EDM-graphs

Also available at http://amc-journaleu ISSN 8-966 printed edn, ISSN 8-97 electronic edn ARS MATHEMATICA CONTEMPORANEA 9 0 6 On minimal forbidden subgraphs for the class of EDM-graphs Gašper Jaklič FMF

### The Geometric Structure of Spanning Trees and Applications to Multiobjective Optimization

The Geometric Structure of Spanning Trees and Applications to Multiobjective Optimization By ALLISON KELLY O HAIR SENIOR THESIS Submitted in partial satisfaction of the requirements for Highest Honors

### Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.

Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.

### Generalizing the Ramsey Problem through Diameter

Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted: January 8, 001; Accepted: November 13, 001. MR Subject Classifications: 05C1, 05C15, 05C35, 05C55 Abstract Given a graph G and positive