# CHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CHAPTER 2 Graphs 1. Introduction to Graphs and Graph Isomorphism 1.1. The Graph Menagerie. Definition A simple graph G = (V, E) consists of a set V of vertices and a set E of edges, represented by unordered pairs of elements of V. A multigraph consists of a set V of vertices, a set E of edges, and a function f : E {{u, v} : u, v V and u v}. If e 1, e 2 E are such that f(e 1 ) = f(e 2 ), then we say e 1 and e 2 are multiple or parallel edges. A pseudograph consists of a set V of vertices, a set E of edges, and a function f : E {{u, v} : u, v V }. If e E is such that f(e) = {u, u} = {u}, then we say e is a loop. A directed graph or digraph G = (V, E) consists of a set V of vertices and a set E of directed edges, represented by ordered pairs of vertices. In Section 1.1 we recall the definitions of the various types of graphs that were introduced in MAD In this section we will revisit some of the ways in which graphs can be represented and discuss in more detail the concept of a graph isomorphism Representing Graphs and Graph Isomorphism. Definition The adjacency matrix, A = [a ij ], for a simple graph G = (V, E), where V = {v 1, v 2,..., v n }, is defined by { 1 if {vi, v a ij = j } is an edge of G, 0 otherwise. 58

2 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 59 We introduce some alternate representations, which are extensions of connection matrices we have seen before, and learn to use them to help identify isomorphic graphs. Remarks Here are some properties of the adjacency matrix of an undirected graph. (1) The adjacency matrix is always symmetric. (2) The vertices must be ordered: and the adjacency matrix depends on the order chosen. (3) An adjacency matrix can be defined for multigraphs by defining a ij to be the number of edges between vertices i and j. (4) If there is a natural order on the set of vertices we will use that order unless otherwise indicated. v 1 v 2 v 3 v 5 v 4 Example An adjacency matrix for this graph is As with connection matrices, an adjacency matrix can be constructed by using a table with the columns and rows labeled with the elements of the vertex set. Here is another example

3 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 60 Example The adjacency matrix for the graph u1 u2 u 5 u 3 u is the matrix M = Incidence Matrices. Definition The incidence matrix, A = [a ij ], for the undirected graph G = (V, E) is defined by { 1 if edge j is incident with vertex i a ij = 0 otherwise. The incidence matrix is another way to use matrices to represent a graph. Remarks (1) This method requires the edges and vertices to be labeled and the matrix depends on the order in which they are written. (2) Every column will have exactly two 1 s. (3) As with adjacency matrices, if there is a natural order for the vertices and edges that order will be used unless otherwise specified Example

4 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 61 v 1 e 1 v 2 e 8 e 2 e 5 e 6 v 3 e 7 e 3 v 5 e 4 v 4 Example The incidence matrix for this graph is Again you can use a table to get the matrix. List all the vertices as the labels for the rows and all the edges for the labels of the columns Degree. Definition (1) Let G = (V, E) be an undirected graph. Two vertices u, v V are adjacent or neighbors if there is an edge e between u and v. The edge e connects u and v. The vertices u and v are endpoints of e. The degree of a vertex v, denoted deg(v), is the number of edges for which it is an endpoint. A loop contributes twice in an undirected graph. If deg(v) = 0, then v is called isolated. If deg(v) = 1, then v is called pendant. (2) Let G = (V, E) be a directed graph. Let (u, v) be an edge in G. Then u is an initial vertex and is adjacent to v. The vertex v is a terminal vertex and is adjacent from u. The in degree of a vertex v, denoted deg (v) is the number of edges which terminate at v.

5 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 62 Similarly, the out degree of v, denoted deg + (v), is the number of edges which initiate at v. We now recall from MAD 2104 the terminology we use with undirected and directed graphs. Notice that a loop contributes two to the degree of a vertex The Handshaking Theorem. Theorem (The Handshaking Theorem). Let G = (V, E) be an undirected graph. Then 2 E = v V deg(v) Proof. Each edge contributes twice to the sum of the degrees of all vertices. The handshaking theorem is one of the most basic and useful combinatorial formulas associated to a graph. It lets us conclude some facts about the numbers of vertices and the possible degrees of the vertices. Notice the immediate corollary. Corollary The sum of the degrees of the vertices in any graph must be an even number. In other words, it is impossible to create a graph so that the sum of the degrees of its vertices is odd (try it!) Example Example Suppose a graph has 5 vertices. Can each vertex have degree 3? degree 4? The sum of the degrees of the vertices would be 3 5 if the graph has 5 vertices of degree 3. This is an odd number, though, so this is not possible by the handshaking Theorem. The sum of the degrees of the vertices if there are 5 vertices with degree 4 is 20. Since this is even it is possible for this to equal 2 E.

6 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 63 If the sum of the degrees of the vertices is an even number then the handshaking theorem is not contradicted. In fact, you can create a graph with any even degree you want if multiple edges are permitted. However, if you add more restrictions it may not be possible. Here are two typical questions the handshaking theorem may help you answer. Exercise Is it possible to have a graph S with 5 vertices, each with degree 4, and 8 edges? Exercise A graph with 21 edges has 7 vertices of degree 1, three of degree 2, seven of degree 3, and the rest of degree 4. How many vertices does it have? 1.8. Theorem Theorem Every graph has an even number of vertices of odd degree. Proof. Let V o be the set of vertices of odd degree, and let V e be the set of vertices of even degree. Since V = V o V e and V o V e =, the handshaking theorem gives us 2 E = deg(v) = deg(v) + deg(v) v V v V o v V e or deg(v) = 2 E deg(v). v V o v V e Since the sum of any number of even integers is again an even integer, the righthand-side of this equations is an even integer. So the left-hand-side, which is the sum of a collection of odd integers, must also be even. The only way this can happen, however, is for there to be an even number of odd integers in the collection. That is, the number of vertices in V o must be even. Theorem goes a bit further than our initial corollary of the handshaking theorem. If you have a problem with the last sentence of the proof, consider the following facts: odd + odd = even odd + even = odd even + even = even If we add up an odd number of odd numbers the previous facts will imply we get an odd number. Thus to get an even number out of v V o deg(v) there must be an even number of vertices in V o (the set of vertices of odd degree).

7 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 64 While there must be an even number of vertices of odd degree, there is no restrictions on the parity (even or odd) of the number of vertices of even degree. This theorem makes it easy to see, for example, that it is not possible to have a graph with 3 vertices each of degree 1 and no other vertices of odd degree Handshaking Theorem for Directed Graphs. Theorem For any directed graph G = (E, V ), E = deg (v) = deg + (v). v V v V When considering directed graphs we differentiate between the number of edges going into a vertex verses the number of edges coming out from the vertex. These numbers are given by the in degree and the out degree. Notice that each edge contributes one to the in degree of some vertex and one to the out degree of some vertex. This is essentially the proof of Theorem Graph Invariants. The following are invariants under isomorphism of a graph G: (1) G has r vertices. (2) G has s edges. (3) G has degree sequence (d 1, d 2,..., d n ). (4) G is a bipartite graph. (5) G contains r complete graphs K n (as a subgraphs). (6) G contains r complete bipartite graphs K m,n. (7) G contains r n-cycles. (8) G contains r n-wheels. (9) G contains r n-cubes. Recall that two simple graphs G 1 = (V 1, E 1 ) and G 2 = (V 2, E 2 ) are isomorphic if there is a bijection f : V 1 V 2 such that vertices u and v in V 1 are adjacent in G 1 if and only if f(u) and f(v) are adjacent in G 2. If there is such a function, we say f is an isomorphism and we write G 1 G 2.

8 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 65 It is often easier to determine when two graphs are not isomorphic. This is sometimes made possible by comparing invariants of the two graphs to see if they are different. We say a property of graphs is a graph invariant (or, just invariant) if, whenever a graph G has the property, any graph isomorphic to G also has the property. The degree sequence a graph G with n vertices is the sequence (d 1, d 2,..., d n ), where d 1, d 2,..., d n are the degrees of the vertices of G and d 1 d 2 d n. Note that a graph could conceivably have infinitely many vertices. If the vertices are countable then the degree sequence would be an infinite sequence. If the vertices are not countable, then this degree sequence would not be defined. The invariants in Section 1.10 may help us determine fairly quickly in some examples that two graphs are not isomorphic. Example Show that the following two graphs are not isomorphic. 1 2 a b 5 6 e f 7 8 g h 3 4 c d G G 2 1 The two graphs have the same number of vertices, the same number of edges, and same degree sequences (3, 3, 3, 3, 2, 2, 2, 2). Perhaps the easiest way to see that they are not isomorphic is to observe that G 1 has three 4-cycles, whereas G 2 has two 4-cycles. In fact, the four vertices of G 1 of degree 3 lie in a 4-cycle in G 1, but the four vertices of G 2 of degree 3 do not. Either of these two discrepancies is enough to show that the graphs are not isomorphic. Another way we could recognize the graphs above are not isomorphic is to consider the adjacency relationships. Notice in G 1 all the vertices of degree 3 are adjacent to 2 vertices of degree 3 and 1 of degree 2. However, in graph G 2 all of the vertices of degree 3 are adjacent to 1 vertex of degree 3 and 2 vertices of degree 2. This discrepancy indicates the two graphs cannot be isomorphic. Example The following two graphs are not isomorphic. Can you find an invariant that is different on the graphs.

9 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM a b 5 6 e f 7 8 g h 3 4 c d G G Example Example Determine whether the graphs G 1 and G 2 are isomorphic. v 1 v 2 u1 u2 v 3 u 5 u 3 v 5 G 1 v 4 G 2 u 4 Solution We go through the following checklist that might tell us immediately if the two are not isomorphic. They have the same number of vertices, 5. They have the same number of edges, 8. They have the same degree sequence (4, 4, 3, 3, 2). Since there is no obvious reason to think they are not isomorphic, we try to construct an isomorphism, f. Note that the above does not tell us there is an isomorphism, only that there might be one.

10 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 67 The only vertex on each that have degree 2 are v 3 f(v 3 ) = u 2. and u 2, so we must have Now, since deg(v 1 ) = deg(v 5 ) = deg(u 1 ) = deg(u 4 ), we must have either f(v 1 ) = u 1 and f(v 5 ) = u 4, or f(v 1 ) = u 4 and f(v 5 ) = u 1. It is possible only one choice would work or both choices may work (or neither choice may work, which would tell us there is no isomorphism). We try f(v 1 ) = u 1 and f(v 5 ) = u 4. Similarly we have two choices with the remaining vertices and try f(v 2 ) = u 3 and f(v 4 ) = u 5. This defines a bijection from the vertices of G 1 to the vertices of G 2. We still need to check that adjacent vertices in G 1 are mapped to adjacent vertices in G 2. To check this we will look at the adjacency matrices. The adjacency matrix for G 1 (when we list the vetices of G 1 by v 1, v 2, v 3, v 4, v 5 ) is A = We create an adjacency matrix for G 2, using the bijection f as follows: since f(v 1 ) = u 1, f(v 2 ) = u 3, f(v 3 ) = u 2, f(v 4 ) = u 5, and f(v 5 ) = u 4, we rearrange the order of the vertices of G 2 to u 1, u 3, u 2, u 5, u 4. With this ordering, the adjacency matrix for G 2 is B = Since A = B, adjacency is preserved under this bijection. Hence the graphs are isomorphic. In this example we show that two graphs are isomorphic. Notice that it is not enough to show they have the same number of vertices, edges, and degree sequence. In fact, if we knew they were isomorphic and we were asked to prove it, we would

11 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 68 proceed directly to try and find a bijection that preserves adjacency. That is, the check list is not necessary if you already know they are isomorphic. On the other hand, having found a bijection between two graphs that doesn t preserve adjacency doesn t tell us the graphs are not isomorphic, because some other bijection might work. If we go down this path, we would have to show that every bijection fails to preserve adjacency. The advantage of the checklist is that it will give you a quick and easy way to show two graphs are not isomorphic if some invariant of the graphs turn out to be different. If you examine the logic, however, you will see that if two graphs have all of the same invariants we have listed so far, we still wouldn t have a proof that they are isomorphic. Indeed, there is no known list of invariants that can be efficiently checked to determine when two graphs are isomorphic. The best algorithms known to date for determining graph isomorphism have exponential complexity (in the number n of vertices). Exercise Determine whether the following two graphs are isomorphic. 1 2 a b 5 6 e f 3 4 c d G G 2 1 Exercise How many different isomorphism (that is, bijections that preserve adjacencies) are possible from G 2 to itself in Example Exercise There are 14 nonisomorphic pseudographs with 3 vertices and 3 edges. Draw all of them. Exercise Draw all nonisomorphic simple graphs with 6 vertices, 5 edges, and no cycles. Exercise Recall the equivalence relation on a set, S, of graphs given by G 1 is related to G 2 if and only if G 1 G 2. How many equivalence classes are there if S is the set of all simple graphs with 6 vertices, 5 edges, and no cycles? Explain Proof of Section 1.10 Part 3 for simple graphs. Proof. Let G 1 and G 2 be isomorphic simple graphs having degree sequences. By part 1 of Section 1.10 the degree sequences of G 1 and G 2 have the same number

12 1. INTRODUCTION TO GRAPHS AND GRAPH ISOMORPHISM 69 of elements (finite or infinite). Let f : V (G 1 ) V (G 2 ) be an isomorphism and let v V (G 1 ). We claim deg G1 (v) = deg G2 (f(v)). If we show this, then f defines a bijection between the vertices of G 1 and G 2 that maps vertices to vertices of the same degree. This will imply the degree sequences are the same. Proof of claim: Suppose deg G1 (v) = k. Then there are k vertices adjacent to v, say u 1, u 2,..., u k. The isomorphism maps each of the vertices to k distinct vertices adjacent to f(u) in G 2 since the isomorphism is a bijection and preserves adjacency. Moreover, f(u) will not be adjacent to any vertices other than the k vertices f(u 1 ), f(u 2 ),... f(u k ). Otherwise, u would be adjacent to the preimage of such a vertex and this preimage would not be one of the vertices u 1, u 2,..., u k since f is an isomorphism. This would contradict that the degree of u is k. This shows the degree of f(u) in G 2 must be k as well, proving our claim. Exercise Prove the remaining properties listed in Section 1.10 for simple graphs using only the properties listed before each and the definition of isomorphism.

### 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

### 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.

### 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

### 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

### 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

### 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

### . 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

### 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

### HOMEWORK #3 SOLUTIONS - MATH 3260

HOMEWORK #3 SOLUTIONS - MATH 3260 ASSIGNED: FEBRUARY 26, 2003 DUE: MARCH 12, 2003 AT 2:30PM (1) Show either that each of the following graphs are planar by drawing them in a way that the vertices do not

### 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

### 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,

### 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.

### 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

### 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

### Graph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick

Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements

### (a) (b) (c) Figure 1 : Graphs, multigraphs and digraphs. If the vertices of the leftmost figure are labelled {1, 2, 3, 4} in clockwise order from

4 Graph Theory Throughout these notes, a graph G is a pair (V, E) where V is a set and E is a set of unordered pairs of elements of V. The elements of V are called vertices and the elements of E are called

### (Vertex) Colorings. We can properly color W 6 with. colors and no fewer. Of interest: What is the fewest colors necessary to properly color G?

Vertex Coloring 2.1 33 (Vertex) Colorings Definition: A coloring of a graph G is a labeling of the vertices of G with colors. [Technically, it is a function f : V (G) {1, 2,...,l}.] Definition: A proper

### CSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph

Today s Topics: CSE 0: Discrete Mathematics for Computer Science Prof. Miles Jones. Graphs. Some theorems on graphs. Eulerian graphs Graphs! Model relations between pairs of objects The Internet graph!

### 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

### 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

### 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

### 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.,

### 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

### 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

### 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.

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

### 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

### 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

### 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

### 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.

### 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).

### 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

### 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?

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### 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

### 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

### 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

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### 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

### Planar Graphs and Graph Coloring

Planar Graphs and Graph Coloring Margaret M. Fleck 1 December 2010 These notes cover facts about graph colorings and planar graphs (sections 9.7 and 9.8 of Rosen) 1 Introduction So far, we ve been looking

### 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

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

### 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

### 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

### Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

### 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

### 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

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### 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

### Chapter 6 Planarity. Section 6.1 Euler s Formula

Chapter 6 Planarity Section 6.1 Euler s Formula In Chapter 1 we introduced the puzzle of the three houses and the three utilities. The problem was to determine if we could connect each of the three utilities

### 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

### 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

### Problem Set. Problem Set #2. Math 5322, Fall December 3, 2001 ANSWERS

Problem Set Problem Set #2 Math 5322, Fall 2001 December 3, 2001 ANSWERS i Problem 1. [Problem 18, page 32] Let A P(X) be an algebra, A σ the collection of countable unions of sets in A, and A σδ the collection

### 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

### diffy boxes (iterations of the ducci four number game) 1

diffy boxes (iterations of the ducci four number game) 1 Peter Trapa September 27, 2006 Begin at the beginning and go on till you come to the end: then stop. Lewis Carroll Consider the following game.

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### Counting Spanning Trees

Counting Spanning Trees Bang Ye Wu Kun-Mao Chao Counting Spanning Trees This book provides a comprehensive introduction to the modern study of spanning trees. A spanning tree for a graph G is a subgraph

### 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.

### 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

### 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

### 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

### 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

### Announcements. CompSci 230 Discrete Math for Computer Science. Test 1

CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 4-7 Prof. Rodger is out Sep 30-Oct 4 There is Recitation: Sept 27-30.

### Determining If Two Graphs Are Isomorphic 1

Determining If Two Graphs Are Isomorphic 1 Given two graphs, it is often really hard to tell if they ARE isomorphic, but usually easier to see if they ARE NOT isomorphic. Here is our first idea to help

### 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

### THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed

### 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

### 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

### Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.

Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection

### 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:

### 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

### 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

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

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

### Characterizations of Arboricity of Graphs

Characterizations of Arboricity of Graphs Ruth Haas Smith College Northampton, MA USA Abstract The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs

### Sets and Subsets. Countable and Uncountable

Sets and Subsets Countable and Uncountable Reading Appendix A Section A.6.8 Pages 788-792 BIG IDEAS Themes 1. There exist functions that cannot be computed in Java or any other computer language. 2. There

### Lecture 4: The Chromatic Number

Introduction to Graph Theory Instructor: Padraic Bartlett Lecture 4: The Chromatic Number Week 1 Mathcamp 2011 In our discussion of bipartite graphs, we mentioned that one way to classify bipartite graphs

### Product irregularity strength of certain graphs

Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### CHAPTER 2 GRAPHS F G C D

page 1 of Section 2.1 HPTR 2 GRPHS STION 1 INTROUTION basic terminology graph is a set of finitely many points called vertices which may be connected by edges. igs 1 3 show three assorted graphs. v1 v2

### 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

### 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

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### Sets and functions. {x R : x > 0}.

Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be expressible in terms of functions.

### Trees and Fundamental Circuits

Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered

### 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. Consider a graph, G in Fig Then the vertex V and edge E can be represented as:

Graph A graph G consist of 1. Set of vertices V (called nodes), (V = {v1, v2, v3, v4...}) and 2. Set of edges E (i.e., E {e1, e2, e3...cm} A graph can be represents as G = (V, E), where V is a finite and

### 1. Determine all real numbers a, b, c, d that satisfy the following system of equations.

altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c

### INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS

INTRODUCTION TO PROOFS: HOMEWORK SOLUTIONS STEVEN HEILMAN Contents 1. Homework 1 1 2. Homework 2 6 3. Homework 3 10 4. Homework 4 16 5. Homework 5 19 6. Homework 6 21 7. Homework 7 25 8. Homework 8 28

### 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

### 6. GRAPH AND MAP COLOURING

6. GRPH ND MP COLOURING 6.1. Graph Colouring Imagine the task of designing a school timetable. If we ignore the complications of having to find rooms and teachers for the classes we could propose the following

### 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

### 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

### 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

### 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

### Mathematical Induction

Mathematical Induction MAT30 Discrete Mathematics Fall 016 MAT30 (Discrete Math) Mathematical Induction Fall 016 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)