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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 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 G = (V, E) is a structure consisting of a set V of vertices (also called nodes), and a set E of edges, which are lines joining vertices. One way to denote an edge e is to explicit the 2 vertices that are connected by this edge, say if the edge e links the vertex u to the vertex v, we write e = {u, v}. Definition 73. Two vertices u, v in a graph G are adjacent in G if {u, v} is an edge of G. If e = {u, v} is an edge of G, then e is called incident with the vertices u and v. Graphs can be of several types. Definition 74. A simple graph G is a graph that has no loop, that is no edge {u, v} with u = v and no parallel edges between any pair of vertices. Example 106. Consider the table of fictitious flights among different cities. Suppose all you want to know is whether there is a direct flight between any two cities, and you are not interested in the direction of the flight. Then you can draw a graph whose vertices are the cities, and there is an edge between city A and city B exactly when there is at least one flight going either from city A to the city B, or from city B to city A. 251

2 252 CHAPTER 11. GRAPH THEORY Definitions A graph G = (V,E) is a structure consisting of a set V of vertices (nodes) and a set E of edges (lines joining vertices). G u e={u,v} v Two vertices u and v are adjacent in G if {u,v} is an edge of G. If e = {u,v}, the edge e is called incident with the vertices u and v. Graphs are useful to represent data. Simple Graphs A simple graph is a graph that has no loop (=edge {u,v} with u=v) and no parallel edges between any pair of vertices. From\To Hong Kong Singapore Beijing Taipei London London Beijing Hong Kong Hong Kong 4 Flights Singapore 2 Flights 3 Flights 1 Flight 1 Flight Taipei Beijing 1 Flight 2 Flights Taipei London 1 Flight 1 Flight 1 Flight Draw a graph to see whether there are direct flights between any two cities (in either direction) Singapore

3 253 Take for example Hong Kong: there will be one edge between Hong Kong and Singapore (this is read in the first row), and and one edge from Hong Kong to Beijing (this is read in the first column). This graph is simple, there is no loop, and no parallel edge. Definition 75. A multigraph G is a graph that has no loop and at least two parallel edges between some pair of vertices. Example 107. We continue with our fictitious example of flights. Suppose now we want to know how many flights are there that operate between two cities (we are still not interested in the direction). In this case, we will need parallel edges to represent multiple flights. For example, let us consider Hong Kong and Singapore: there are 4 flights from Hong Kong to Singapore, and 2 flights from Singapore to Hong Kong, thus a total of 6 edges between the two vertices representing these cities. Very often, a relation from a vertex A to B does not yield one from B to A, in this case, edges become arrows. Definition 76. A directed graph G, also called digraph for short, is a graph where edges {u, v} are ordered ({u, v} and {v, u} are not the same), that is edges have a direction. The graph is called undirected otherwise. Parallel edges are allowed in directed multigraph. Loops are allowed for both directed and directed multigraphs. Example 108. On our example of ficticious flights, a directed graph corresponds to put arrows for flights going in a given direction, for example, there is one arrow from London to Beijing, but none from Beijing to London. People attribute the origin of graph theory to the Königsberg bridge question, solved by the mathematician Euler in The question was, given the map of Königsberg, which contains 7 bridges, is it possible to find a walk that goes through the 7 bridges without crossing a bridge twice? You may look at the map and give it a try yourself, to convince yourself of the answer. The answer turns out to be no, it is not possible. We will see why next. But first, we will give a name to such walks, in honour of Euler: Definition 77. A Euler path/trail is a walk on the edges of a graph which uses each edge in the graph exactly once. A Euler circuit/cycle is a walk on the edges of a graph which starts and ends at the same vertex, and uses each edge in the graph exactly once.

4 254 CHAPTER 11. GRAPH THEORY Multigraphs A multigraph is a graph that has no loop and at least 2 parallel edges between some pair of vertices. From\To Hong Kong Singapore Beijing Taipei London London Beijing Hong Kong Hong Kong 4 Flights Singapore 2 Flights 3 Flights 1 Flight 1 Flight Taipei Beijing 1 Flight 2 Flights Taipei London 1 Flight 1 Flight 1 Flight Draw a graph with an edge for each flight that operates between two cities (in either direction). Singapore 4/15 Directed (Multi)graphs A directed graph is a graph where edges {u,v} are ordered, that is, edges have a direction. Parallel edges are allowed in directed multigraphs. Loops are allowed for both. From\To Hong Kong Singapore Beijing Taipei London Hong Kong 4 Flights Singapore 2 Flights 3 Flights 1 Flight 1 Flight Beijing 1 Flight 2 Flights Taipei London Hong Kong Taip ei Beijing London 1 Flight 1 Flight 1 Flight Draw a graph to see whether there are direct flights between any two cities (direction matters) Singapore

5 255 Origin: The Bridges of Konigsberg Königsberg (now Kaliningrad, Russia) has 7 bridges. a d b c People tried (without success) to find a way to walk all 7 bridges without crossing a bridge twice. b a c d Leonhard Euler introduced Graphs in 1736 to solve the Königsberg Bridge problem (no solution and why). Euler Path and Circuit A Euler path (Eulerian trail) is a walk on the edges of a graph which uses each edge in the original graph exactly once. The beginning and end of the walk may or not be the same vertex. A Euler circuit (Eulerian cycle) is a walk on the edges of a graph which starts and ends at the same vertex, and uses each edge in the original graph exactly once.

6 256 CHAPTER 11. GRAPH THEORY Euler Circuit b a c d Suppose the beginning and end are the same node u. The graph must be connected. At every vertex v u, we reach v along one edge and go out along another, thus the number of edges incident at v (called the degree of v) is even. The node u is visited once the first time we leave, and once the last time we arrive, and possibly in between (back and forth), thus the degree of u is even. Since the Königsberg Bridges graph has odd degrees, no solution! 8/15 Euler Theorem The degree of a vertex is the number of edges incident with it. Theorem. Consider a connected graph G. 1. If G contains an Euler path that starts and ends at the same node, then all nodes of G have an even degree. 2. If G contains an Euler path, then exactly two nodes of G have an odd degree. Suppose G as an Euler path, which starts at v and finishes at w. Add the edge {v,w}. Then by the first part of the theorem, all nodes have even degree, but for v and w which have odd degrees.

7 257 The Euler circuit/cycle is simply an Euler path/trail whose start and end are the same vertex. Definition 78. The degree of a node is the number of edges incident with it. With this definition of degree, we next answer the question of the bridges of K nigsberg. We start with the more constrained case of starting and finishing at the same vertex. We assume the graph G is connected, which means that there is always a way to walk from any vertex to any other (possibly using several times the same vertex or the same edge). Theorem 4. Consider a connected graph G. Then G contains an Euler circuit/cycle, if and only if all nodes of G have an even degree. Proof. We prove only that if G contains an Euler circuit, then all nodes have an even degree. We start the walk at vertex u. Now for any vertex v which is not u, we need to walk in v using some edge, and walk out of u using another edge. We may come back to v, but for every come back, we still need one edge to come in, and one to walk out. Therefore the degree of v must be even! As for the starting point u, it is visited once the first time we leave, and the last time we arrive (2 edges), and any possible back and forth counts for 2 edges as well, which shows that indeed, it must be that all nodes have an even degree! Since the Königsberg bridge graph has odd degrees, it has no Euler cycle. We next extend the argument to an Euler path. Theorem 5. Consider a connected graph G. Then G contains an Euler path if and only if exactly two nodes of G have an odd degree. Proof. We only prove that if G has an Euler path, then exactly two nodes of G have an odd degree. Suppose thus that G has an Euler path, which starts at v and finishes at w. Create a new graph G, which is formed from G by adding one edge between v and w. Now G has an Euler cycle, and so we know by the previoust theorem that G has the property that all its vertices have an even degree. Therefore the degrees of v and w in G are odd, while all the others are even and we are done. The other direction is left as an exercise (see Exercise 96).

8 Boat River 258 CHAPTER 11. GRAPH THEORY Examples Note: Euler Theorem actually states an if and only if. Example: Wolf, Goat, Cabbage A classical puzzle that involves graphs: Ferryman From the left bank of the river, the ferryman is to transport the wolf, the goat and the cabbage to the right bank. The boat is only big enough to transport one object/animal at a time, other than himself. The wolf cannot be left alone with the goat, and the goat cannot be left alone with the cabbage. How should the ferryman proceed? Copyright belongs to the artists 11/15

9 259 Here is a classical puzzle. Example 109. A ferryman needs to transport a wolf, a goat and a cabbage from one side of a river to the other, the boat is big enough for himself and one object/animal at a time. How should the ferryman proceed, knowing that the wolf cannot be left alone with the goat, and the goat cannot be left alone with the cabbage? One way to solve this is to use a graph that represents the different possible states of this sytem. The initial start up point is a state where wgcf (w=wolf, g=goat, c=cabbage and f =ferryman) are on the left side of the river, with nothing on the right side. The first step, the ferryman has no choice, he takes the goat on the other side, which leads to a second step, with wc on the leftt bank, and gf on the right bank. The ferryman returns, he then can choose: he either takes the cabbage of the wolf. This creates two branches in the graph. Each branch leads to a couple of states, after which (see the graph itself) we reach a state where g is on the left, and wfc is on the right, leading to the end of the puzzle. A solution is a path in this graph that represents the different states of the system. In this example, it is a fairly easy graph, and there are 2 paths, each of the same length! Here are two more types of graphs which are important, in that you are very likely to encounter them. Definition 79. A complete graph with n vertices is a simple graph that has every vertex connected to every other distinct vertex. Definition 80. A bipartite graph is a graph whose vertices can be partitioned into 2 (disjoint) subset V and W such that each edge only connects a v V and a w W. We have seen the notion of degree of a vertex v in an undirected graph, it is the number of edges incident with. We note that in this case, a loop at a vertex contributes twice. For directed graphs, you may like to know that the notion of degree is more precise, one distinguishes in-degree and out-degree, which we will not discussed here. Definition 81. The total degree of an undirected graph G = (V, E) is the sum of the degrees of all the vertices of G: v V deg(v).

10 260 CHAPTER 11. GRAPH THEORY Example: Wolf, Goat, Cabbage (IV) f = ferryman g = goat w = wolf c = cabbage Either he takes 1. The cabbage, bring back the goat, leave the goat and take the wolf across, return, and take the goat across. 2. Or the wolf, bring back the goat, leave the goat and the cabbage across, return, and take the goat across. w/cfg wgf/c wgcf/ wc/gf wcf/g g/wfc gf/wc /wcfg c/wfg cfg/w Complete & Bipartite A complete graph with n vertices is a simple graph that has every vertex connected to every other distinct vertex. A bipartite graph is a graph whose vertices can be partitioned into 2 (disjoint) subsets V and W s.t. each edge only connects a v V and a w W.

11 261 More on Node Degree The degree deg(v) of a vertex v in an undirected graph is the number of edges incident with it ( a loop at a vertex contributes twice). In-degree and Out-degree are distinguished for directed graphs. v 2 total degree = deg(v 1 ) + deg(v 2 ) + deg(v 3 ) = = 6. v 1 v 3 The total degree deg(g) of an undirected graph G is the sum of the degrees of all the vertices of G: v V deg(v) The Handshaking Theorem Let G = (V,E) be an undirected graph with e edges. Then 2e = v V deg(v) (Note that this even applies if multiple edges and loops are present.) v 1 v 2 Proof. Choose an e E(G) with endpoints v, w V. e contributes 1 to deg(v) and 1 to deg(w). True even when v = w. Thus each edge contributes 2 to the total degree. v 3 v 4 deg(v 1 ) = deg(v 2 ) = deg(v 3 ) = deg(v 4 ) = 4 2e = deg(v) = 4 4 = 16 and e = 8

12 262 CHAPTER 11. GRAPH THEORY Adjacency Matrix v 1 v 2 v 3 v v 1 v 2 A = v 2 v v 3 A graph can be represented by a matrix A = (a ij ) called adjacency matrix, with a ij = the number of arrows from v i to v j What is the adjacency matrix of a complete graph? Hamiltonian Circuit The Icosian game (1857): Along the edges of a dodecahedron, find a path such that every vertex is visited a single time, and the ending point is the same as the starting point. Hamilton sold it to a London game dealer in 1859 for 25 pounds. A Hamiltonian path of a graph G is a walk such that every vertex is visited exactly once. A Hamiltonian circuit of a graph G is a closed walk such that every vertex is visited exactly once (except the same start/end vertex). Copyright:

13 263 The Handshaking Theorem links the number of egdes in a graph to the numbers of vertices. Theorem 6. Let G = (V, E) be an undirected graph with E edges. Then 2 E = v V deg(v). Proof. The proof follows the name of the theorem. The idea of handshaking is that if two people shake hands, there must be...well...two persons involved. In a graph G, this becomes, if there is an edge e between v and w, then e contributes to 1 to the degree of v and to 1 to the degree of s. This is also true when v = w. Therefore each edge contributes 2 to the total degree. To represent a graph, a useful way to do so is to use a matrix. Definition 82. The adjacency matrix of a graph G is a matrix A whose coefficients are denoted a ij, where a ij, the coefficient in the ith row and jth column, counts the number of arrows from v i to v j. You may replace the term arrow by edge in this definition if you graph is undirected. Example 110. The adjacency of a complete graph with 4 vertices is The first row reads that v 1 is connected to v 2, v 3, v 4, but not to itself (since it is a simple graph by definition). We saw earlier Euler paths as walks going through exactly every edge of a graph. If you replace exactly every edge by exactly every vertex, this becomes a Hamiltonian path! Definition 83. A Hamiltonian path of a graph G is a walk such that every vertex is visited exactly one. A Hamiltonian circuit of a graph G is a closed walk such that every vertex is visited exactly one, except the same start/end vertex. Hamiltonian paths are harder to characterize than Euler paths. In particular, finding an algorithm that will identify Hamiltonian paths in a graph is hard!

14 264 CHAPTER 11. GRAPH THEORY Hamiltonian vs Eulerian Path (or trail) vs Circuit (or cycle): for circuits, the walk starts and finishes at the same vertex, not needed for path. Eulerian: walk through every edge exactly once. Hamiltonian: walk through every vertex exactly. Examples Euler circuit Hamiltonian path No Hamiltonian circuit No Euler circuit, but Euler path Hamiltonian path No Hamiltonian circuit

15 265 Graph Isomorphism v 1 v 3 v 4 v 1 v 5 v v 5 v 2 v 4 v 3 Finally, it is useful to pay attention that one draws a graph, it is just a visualization...and several visualizations may be different, giving the impression that the graphs are different, while they are actually the same. If two graphs differ by their labeling, but their adjacency structure is the same, we say that these graphs are isomorphic. More formally: Definition 84. A graph isomorphism between two graphs G and H is a bijection f between the set of vertices of G and the set of vertices of H such that any two vertices u, v in G are adjacent if and only if f(u), f(v) are adjacent in H.

16 266 CHAPTER 11. GRAPH THEORY Exercises for Chapter 11 Exercise 96. Prove that if a connected graph G has exactly two vertices which have odd degree, then it contains an Euler path. Exercise 97. Draw a complete graph with 5 vertices. Exercise 98. Show that in every graph G, the number of vertices of odd degree is even. Exercise 99. Show that in very simple graph (with at least two vertices), there must be two vertices that have the same degree. Exercise 100. Decide whether the following graphs contain a Euler path/cycle.

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

### 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

### 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

### Euler Paths and Euler Circuits

Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

### 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

### Section Summary. Introduction to Graphs Graph Taxonomy Graph Models

Chapter 10 Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity Euler and Hamiltonian Graphs Shortest-Path Problems

### 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

### Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh

Graph Theory Origin and Seven Bridges of Königsberg -Rhishikesh Graph Theory: Graph theory can be defined as the study of graphs; Graphs are mathematical structures used to model pair-wise relations between

### 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

### CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

### 3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

### Introduction to Graph Theory

Introduction to Graph Theory Allen Dickson October 2006 1 The Königsberg Bridge Problem The city of Königsberg was located on the Pregel river in Prussia. The river divided the city into four separate

### Simple Graphs Degrees, Isomorphism, Paths

Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### 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

### 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

### 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

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### Answer: (a) Since we cannot repeat men on the committee, and the order we select them in does not matter, ( )

1. (Chapter 1 supplementary, problem 7): There are 12 men at a dance. (a) In how many ways can eight of them be selected to form a cleanup crew? (b) How many ways are there to pair off eight women at the

### Network/Graph Theory. What is a Network? What is network theory? Graph-based representations. Friendship Network. What makes a problem graph-like?

What is a Network? Network/Graph Theory Network = graph Informally a graph is a set of nodes joined by a set of lines or arrows. 1 1 2 3 2 3 4 5 6 4 5 6 Graph-based representations Representing a problem

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

### 136 CHAPTER 4. INDUCTION, GRAPHS AND TREES

136 TER 4. INDUCTION, GRHS ND TREES 4.3 Graphs In this chapter we introduce a fundamental structural idea of discrete mathematics, that of a graph. Many situations in the applications of discrete mathematics

### 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

### Crossing the Bridge at Night

A1:01 Crossing the Bridge at Night Günter Rote Freie Universität Berlin, Institut für Informatik Takustraße 9, D-14195 Berlin, Germany rote@inf.fu-berlin.de A1:02 August 21, 2002 A1:03 A1:04 Abstract We

### Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1

GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept

### princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora

princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of

### 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

### Formal Languages and Automata Theory - Regular Expressions and Finite Automata -

Formal Languages and Automata Theory - Regular Expressions and Finite Automata - Samarjit Chakraborty Computer Engineering and Networks Laboratory Swiss Federal Institute of Technology (ETH) Zürich March

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive

### Chapter 6: Graph Theory

Chapter 6: Graph Theory Graph theory deals with routing and network problems and if it is possible to find a best route, whether that means the least expensive, least amount of time or the least distance.

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### Graph theory and network analysis. Devika Subramanian Comp 140 Fall 2008

Graph theory and network analysis Devika Subramanian Comp 140 Fall 2008 1 The bridges of Konigsburg Source: Wikipedia The city of Königsberg in Prussia was set on both sides of the Pregel River, and included

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### 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,

### Spring 2007 Math 510 Hints for practice problems

Spring 2007 Math 510 Hints for practice problems Section 1 Imagine a prison consisting of 4 cells arranged like the squares of an -chessboard There are doors between all adjacent cells A prisoner in one

### 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...............................

### 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

### Distributed Computing over Communication Networks: Maximal Independent Set

Distributed Computing over Communication Networks: Maximal Independent Set What is a MIS? MIS An independent set (IS) of an undirected graph is a subset U of nodes such that no two nodes in U are adjacent.

### Sum of Degrees of Vertices Theorem

Sum of Degrees of Vertices Theorem Theorem (Sum of Degrees of Vertices Theorem) Suppose a graph has n vertices with degrees d 1, d 2, d 3,...,d n. Add together all degrees to get a new number d 1 + d 2

### Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902

Graph Theory Lecture 3: Sum of Degrees Formulas, Planar Graphs, and Euler s Theorem Spring 2014 Morgan Schreffler Office: POT 902 http://www.ms.uky.edu/~mschreffler Different Graphs, Similar Properties

### Networks and Paths. The study of networks in mathematics began in the middle 1700 s with a famous puzzle called the Seven Bridges of Konigsburg.

ame: Day: etworks and Paths Try This: For each figure,, and, draw a path that traces every line and curve exactly once, without lifting your pencil.... Figures,, and above are examples of ETWORKS. network

### Special Classes of Divisor Cordial Graphs

International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com

### Asking Hard Graph Questions. Paul Burkhardt. February 3, 2014

Beyond Watson: Predictive Analytics and Big Data U.S. National Security Agency Research Directorate - R6 Technical Report February 3, 2014 300 years before Watson there was Euler! The first (Jeopardy!)

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### Session 6 Number Theory

Key Terms in This Session Session 6 Number Theory Previously Introduced counting numbers factor factor tree prime number New in This Session composite number greatest common factor least common multiple

### Course on Social Network Analysis Graphs and Networks

Course on Social Network Analysis Graphs and Networks Vladimir Batagelj University of Ljubljana Slovenia V. Batagelj: Social Network Analysis / Graphs and Networks 1 Outline 1 Graph...............................

### Quantum Monte Carlo and the negative sign problem

Quantum Monte Carlo and the negative sign problem or how to earn one million dollar Matthias Troyer, ETH Zürich Uwe-Jens Wiese, Universität Bern Complexity of many particle problems Classical 1 particle:

### Elementary Algebra. Section 0.4 Factors

Section 0.4 Contents: Definitions: Multiplication Primes and Composites Rules of Composite Prime Factorization Answers Focus Exercises THE MULTIPLICATION TABLE x 1 2 3 4 5 6 7 8 9 10 11 12 1 1 2 3 4 5

### 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

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### 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

### Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

### A permutation can also be represented by describing its cycles. What do you suppose is meant by this?

Shuffling, Cycles, and Matrices Warm up problem. Eight people stand in a line. From left to right their positions are numbered,,,... 8. The eight people then change places according to THE RULE which directs

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

### Pigeonhole Principle Solutions

Pigeonhole Principle Solutions 1. Show that if we take n + 1 numbers from the set {1, 2,..., 2n}, then some pair of numbers will have no factors in common. Solution: Note that consecutive numbers (such

### v w is orthogonal to both v and w. the three vectors v, w and v w form a right-handed set of vectors.

3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with

### PROBLEM SET 7: PIGEON HOLE PRINCIPLE

PROBLEM SET 7: PIGEON HOLE PRINCIPLE The pigeonhole principle is the following observation: Theorem. Suppose that > kn marbles are distributed over n jars, then one jar will contain at least k + marbles.

### Practical Guide to the Simplex Method of Linear Programming

Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear

### 6.3 Conditional Probability and Independence

222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

### Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

### SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE

SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH NATIONAL UNIVERSITY OF SINGAPORE 2012 SPANNING CACTI FOR STRUCTURALLY CONTROLLABLE NETWORKS NGO THI TU ANH (M.Sc., SFU, Russia) A THESIS

### Kings in Tournaments. Yu Yibo, Di Junwei, Lin Min

Kings in Tournaments by Yu Yibo, i Junwei, Lin Min ABSTRACT. Landau, a mathematical biologist, showed in 1953 that any tournament T always contains a king. A king, however, may not exist. in the. resulting

### A Fast Algorithm For Finding Hamilton Cycles

A Fast Algorithm For Finding Hamilton Cycles by Andrew Chalaturnyk A thesis presented to the University of Manitoba in partial fulfillment of the requirements for the degree of Masters of Science in Computer

### 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

### P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### Any two nodes which are connected by an edge in a graph are called adjacent node.

. iscuss following. Graph graph G consist of a non empty set V called the set of nodes (points, vertices) of the graph, a set which is the set of edges and a mapping from the set of edges to a set of pairs

### Lecture Notes: Matrix Inverse. 1 Inverse Definition

Lecture Notes: Matrix Inverse Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Inverse Definition We use I to represent identity matrices,

### Lecture 1: Course overview, circuits, and formulas

Lecture 1: Course overview, circuits, and formulas Topics in Complexity Theory and Pseudorandomness (Spring 2013) Rutgers University Swastik Kopparty Scribes: John Kim, Ben Lund 1 Course Information Swastik

### 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.

### MATHEMATICAL ENGINEERING TECHNICAL REPORTS. An Improved Approximation Algorithm for the Traveling Tournament Problem

MATHEMATICAL ENGINEERING TECHNICAL REPORTS An Improved Approximation Algorithm for the Traveling Tournament Problem Daisuke YAMAGUCHI, Shinji IMAHORI, Ryuhei MIYASHIRO, Tomomi MATSUI METR 2009 42 September

### Scheduling a Tournament

Scheduling a Tournament Dalibor Froncek University of Minnesota Duluth February 9, 00 Abstract We present several constructions for scheduling round robin tournaments. We explain how to schedule a tournament

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### Induction. Margaret M. Fleck. 10 October These notes cover mathematical induction and recursive definition

Induction Margaret M. Fleck 10 October 011 These notes cover mathematical induction and recursive definition 1 Introduction to induction At the start of the term, we saw the following formula for computing

### Mathematical Induction. Mary Barnes Sue Gordon

Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by

### 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

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### CS311H. Prof: Peter Stone. Department of Computer Science The University of Texas at Austin

CS311H Prof: Department of Computer Science The University of Texas at Austin Good Morning, Colleagues Good Morning, Colleagues Are there any questions? Logistics Class survey Logistics Class survey Homework

### Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way

Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)

### Tree-representation of set families and applications to combinatorial decompositions

Tree-representation of set families and applications to combinatorial decompositions Binh-Minh Bui-Xuan a, Michel Habib b Michaël Rao c a Department of Informatics, University of Bergen, Norway. buixuan@ii.uib.no

### Network Flow I. Lecture 16. 16.1 Overview. 16.2 The Network Flow Problem

Lecture 6 Network Flow I 6. Overview In these next two lectures we are going to talk about an important algorithmic problem called the Network Flow Problem. Network flow is important because it can be

### 5.1 Bipartite Matching

CS787: Advanced Algorithms Lecture 5: Applications of Network Flow In the last lecture, we looked at the problem of finding the maximum flow in a graph, and how it can be efficiently solved using the Ford-Fulkerson

### Mathematics for Algorithm and System Analysis

Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### 3 Some Integer Functions

3 Some Integer Functions A Pair of Fundamental Integer Functions The integer function that is the heart of this section is the modulo function. However, before getting to it, let us look at some very simple

### SOME APPLICATIONS OF EULERIAN GRAPHS

Sutra: International Journal of Mathematical Science Education Technomathematics Research Foundation Vol. 2, No. 2, 1 10, 2009 SOME APPLICATIONS OF EULERIAN GRAPHS Abdul Samad Ismail, Roslan Hasni and

### 1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT

DECISION 1 Revision Notes 1. Sorting (assuming sorting into ascending order) a) BUBBLE SORT Make sure you show comparisons clearly and label each pass First Pass 8 4 3 6 1 4 8 3 6 1 4 3 8 6 1 4 3 6 8 1

### COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### Part 2: Community Detection

Chapter 8: Graph Data Part 2: Community Detection Based on Leskovec, Rajaraman, Ullman 2014: Mining of Massive Datasets Big Data Management and Analytics Outline Community Detection - Social networks -

### 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

### Arrangements And Duality

Arrangements And Duality 3.1 Introduction 3 Point configurations are tbe most basic structure we study in computational geometry. But what about configurations of more complicated shapes? For example,

### 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

### Euler, Mei-Ko Kwan, Königsberg, and a Chinese Postman

Documenta Math. 43 Euler, Mei-Ko Kwan, Königsberg, and a Chinese Postman Martin Grötschel and Ya-xiang Yuan 2010 Mathematics Subject Classification: 00-02, 01A05, 05C38, 90-03 Keywords and Phrases: Eulerian

### 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

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the