1 Introduction to Graph Theory


 Poppy Conley
 10 months ago
 Views:
Transcription
1 CHAPTER 1 Introduction to Graph Theory INTRODUCTION It is no coincidence that graph theory has been independently discovered many times, since it may quite properly be regarded as an area of applied mathematics. The basic combinatorial nature of graph theory and a clue to its wide applicability are indicated in the words of Sylvester, The theory of ramification is one of pure colligation, for it takes no account of magnitude or position ; geometrical lines are used, but have no more real bearing on the matter than those employed in genealogical tables have in explaining the laws of procreation. Indeed, the earliest recorded mention of the subject occurs in the works of Euler, and although the original problem he was considering might be regarded as a somewhat frivolous puzzle, it did arise from the physical world. Subsequent rediscoveries of graph theory by Kirchhoff and Cayley also had their roots in the physical world. Kirchhoff s investigations of electric networks led to his development of the basic concepts and theorems concerning trees in graphs, while Cayley considered trees arising from the enumeration of organic chemical isomers. Another puzzle approach to graphs was proposed by Hamilton. After this, the celebrated four colour conjecture came into prominence and has been notorious ever since. In the present century, there have already been a great many rediscoveries of graph theory which we can only mention most briefly in this chronological account. Euler ( ) became the father of graph theory as well as topology. Graph theory is considered to have begun in 1736 with the publication of Euler s solution of the Königsberg bridge problem. The graph theory is one of the few fields of mathematics with a definite birth date by ore. 1.1 WHAT IS A GRAPH? DEFINITION A graph G consists of a set of objects V = {v 1, v 2, v 3,...} called vertices (also called points or nodes) and other set E = {e 1, e 2, e 3,...} whose elements are called edges (also called lines or arcs). The set V(G) is called the vertex set of G and E(G) is the edge set. Usually the graph is denoted as G = (V, E) Let G be a graph and {u, v} an edge of G. Since {u, v} is 2element set, we may write {v, u} instead of {u, v}. It is often more convenient to represent this edge by uv or vu. If e = uv is an edge of a graph G, then we say that u and v are adjacent in G and that e joins u and v. (We may also say that each that of u and v is adjacent to or with the other). For example : A graph G is defined by the sets V(G) = {u, v, w, x, y, z} and E(G) = {uv, uw, wx, xy, xz}. 1
2 2 GRAPH THEORY WITH APPLICATIONS Now we have the following graph by considering these sets. Every graph has a diagram associated with it. The vertex u and an edge e are incident with each other as are v and e. If two distinct edges say e and f are incident with a common vertex, then they are adjacent edges. A graph with pvertices and qedges is called a (p, q) graph. The (1, 0) graph is called trivial graph. In the following figure the vertices a and b are adjacent but a and c are not. The edges x and y are adjacent but x and z are not. Although the edges x and z intersect in the diagram, their intersection is not a vertex of the graph. Examples : (1) Let V = {1, 2, 3, 4} and E = {{1, 2}, {1, 3}, {3, 2}. {4, 4}}. Then G(V, E) is a graph. (2) Let V = {1, 2, 3, 4} and E = {{1, 5}, {2, 3}}. Then G(V, E) is not a graph, as 5 is not in V. (3) A graph with 5vertices and 8edges is called a (5, 8) graph.
3 INTRODUCTION TO GRAPH THEORY DIRECTED AND UNDIRECTED GRAPHS Directed graph A directed graph or digraph G consists of a set V of vertices and a set E of edges such that e E is associated with an ordered pair of vertices. In other words, if each edge of the graph G has a direction then the graph is called directed graph. In the diagram of directed graph, each edge e = (u, v) is represented by an arrow or directed curve from initial point u of e to the terminal point v. Figure 1(a) is an example of a directed graph. Fig. 1(a). Directed graph. Suppose e = (u, v) is a directed edge in a digraph, then (i) u is called the initial vertex of e and v is the terminal vertex of e (ii) e is said to be incident from u and to be incident to v. (iii) u is adjacent to v, and v is adjacent from u Undirected graph An undirected graph G consists of set V of vertices and a set E of edges such that each edge e E is associated with an unordered pair of vertices. In other words, if each edge of the graph G has no direction then the graph is called undirected graph. Figure 1(b) is an example of an undirected graph. We can refer to an edge joining the vertex pair i and j as either (i, j) or (j, i). Figure 1(b). Undirected graph.
4 4 GRAPH THEORY WITH APPLICATIONS 1.3 BASIC TERMINOLOGIES Loop : An edge of a graph that joins a node to itself is called loop or self loop. i.e., a loop is an edge (v i, v j ) where v i = v f Multigraph In a multigraph no loops are allowed but more than one edge can join two vertices, these edges are called multiple edges or parallel edges and a graph is called multigraph. Two edges (v i, v j ) and (v f, v r ) are parallel edges if v i = v r and v j, v f. Fig. 2(a) Fig. 2(b) In Figure 1.2(a), there are two parallel edges associated with v 2 and v 3. In Figure 1.2(b), there are two parallel edges joining nodes v 1 and v 2 and v 2 and v Pseudo graph A graph in which loops and multiple edges are allowed, is called a pseudo graph. Fig. 3(a) Fig. 3(b) Simple graph A graph which has neither loops nor multiple edges. i.e., where each edge connects two distinct vertices and no two edges connect the same pair of vertices is called a simple graph. Figure 1.1(a) and (b) represents simple undirected and directed graph because the graphs do not contain loops and the edges are all distinct Finite and Infinite graphs A graph with finite number of vertices as well as a finite number of edges is called a finite graph. Otherwise, it is an infinite graph.
5 INTRODUCTION TO GRAPH THEORY DEGREE OF A VERTEX The number of edges incident on a vertex v i with selfloops counted twice (is called the degree of a vertex v i and is denoted by deg G (v i ) or deg v i or d(v i ). The degrees of vertices in the graph G and H are shown in Figure 4(a) and 4(b). Fig. 4(a) Fig. 4(b) In G as shown in Figure 4(a), deg G (v 2 ) = 2 = deg G (v 4 ) = deg G (v 1 ), deg G (v 3 ) = 3 and deg G (v 5 ) = 1 and In H as shown in Figure 4(b), deg H (v 2 ) = 5, deg H (v 4 ) = 3, deg H (v 3 ) = 5, deg H (v 1 ) = 4 and deg H (v 5 ) = 1. The degree of a vertex is some times also referred to as its valency. 1.5 ISOLATED AND PENDENT VERTICES Isolated vertex A vertex having no incident edge is called an isolated vertex. In other words, isolated vertices are those with zero degree Pendent or end vertex A vertex of degree one, is called a pendent vertex or an end vertex. In the above Figure, v 5 is a pendent vertex In degree and out degree In a graph G, the out degree of a vertex v i of G, denoted by out deg G (v i ) or number of edges beginning at v i and the in degree of v i, denoted by in deg G (v i ) or deg G + 1 deg G (v i ), is the (v i ), is the number of edges ending at v i. The sum of the in degree and out degree of a vertex is called the total degree of the vertex. A vertex with zero in degree is called a source and a vertex with zero out degree is called a sink. Since each edge has an initial vertex and terminal vertex. 1.6 THE HANDSHAKING THEOREM 1.1 If G = (v, E) be an undirected graph with e edges. i.e., Then v V deg ( v) = 2e G the sum of degrees of the vertices is an undirected graph is even.
6 6 GRAPH THEORY WITH APPLICATIONS Proof : Since the degree of a vertex is the number of edges incident with that vertex, the sum of the degree counts the total number of times an edge is incident with a vertex. Since every edge is incident with exactly two vertices, each edge gets counted twice, once at each end. Thus the sum of the degrees equal twice the number of edges. Note : This theorem applies even if multiple edges and loops are present. The above theorem holds this rule that if several people shake hands, the total number of hands shake must be even that is why the theorem is called handshaking theorem. Corollary : In a non directed graph, the total number of odd degree vertices is even. Proof : Let G = (V, E) a non directed graph. Let U denote the set of even degree vertices in G and W denote the set of odd degree vertices. Then deg G ( vi ) = G vi V vi U deg ( vi ) + deg G ( vi ) vi W 2e deg G ( v1) = deg G ( v1)...(1) v i U v i W Now deg G ( vi ) is also even vi W Therefore, from (1) deg G ( vi ) is even vi W The no. of odd vertices in G is even. Theorem 1.2. If V = {v 1, v 2,... v n } is the vertex set of a non directed graph G, then n i = 1 deg ( v ) = 2 E i If G is a directed graph, then n n + vi i = 1 i = 1 deg ( ) = deg ( v ) = E Proof : Since when the degrees are summed. Each edge contributes a count of one to the degree of each of the two vertices on which the edge is incident. Corollary (1) : In any non directed graph there is an even number of vertices of odd degree. Proof : Let W be the set of vertices of odd degree and let U be the set of vertices of even degree. Then deg ( v) = deg ( v) + deg ( v) = 2 E v V(G) v W v U i Certainly, deg ( v) is even, v U
7 INTRODUCTION TO GRAPH THEORY 7 Hence deg ( v) is even, v W Implying that W is even. Corollary (2) : If k = δ(g) is the minimum degree of all the vertices of a non directed graph G, then k V v V(G) deg ( v) = 2 E In particular, if G is a kregular graph, then k V = deg ( v) = 2 E. v V(G) Problem 1.1. Show that, in any gathering of six people, there are either three people who all know each other or three people none of whom knows either of the other two (six people at a party). Solution. To solve this problem, we draw a graph in which we represent each person by a vertex and join two vertices by a solid edge if the corresponding people know each other, and by a dotted edge if not. We must show that there is always a solid triangle or a dotted triangle. Let v be any vertex. Then there must be exactly five edges incident with v, either solid or dashed, and so at least three of these edges must be of the same type. Let us assume that there are three solid edges (see figure 5) ; the case of atleast three dashed edges is similar. Fig. 5. If the people corresponding to the vertices w and x know each other, then v, w and x form a solid triangle, as required. Similarly, if the people corresponding to the vertices w and y, or to the vertices x and y, know each other, then we again obtain a solid triangle. These three cases are shown in Figure (6).
8 8 GRAPH THEORY WITH APPLICATIONS Fig. 6. Finally, if no two of the people corresponding to the vertices w, x and y know each other, then w, x and y from a dotted triangle, as required (see figure (7). Fig. 7. Problem 1.2. Place the letters A, B, C, D, E, F, G, H into the eight circles in Figure (8), in such a way that no letter is adjacent to a letter that is next to it in the alphabet. Fig. 8. Solution. First note that trying all the possibilities is not a practical proposition, as there are 8! = ways of placing eight letters into eight circles. Note that (i) the easiest letters to place are A and H, because each has only one letter to which it cannot be adjacent, namely, B and G, respectively. (ii) the hardest circles to fill are those in the middle, as each is adjacent to six others. This suggests that we place A and H in the middle circles. If we place A to the left of H, then the only possible positions for B and G are shown in Figure (9). Fig. 9.
9 INTRODUCTION TO GRAPH THEORY 9 The letter C must now be placed on the lefthand side of the diagram, and F must be placed on the righthand side. It is then a simple matter to place the remaining letters, as shown in Figure (10). Fig. 10. Problem 1.3. Determine the number of edges in a graph with 6 vertices, 2 of degree 4 and 4 of degree 2. Draw two such graphs. Solution. Suppose the graph with 6 vertices has e number of edges. Therefore by Handshaking lemma 6 i = 1 deg ( v ) = 2e i d(v 1 ) + d(v 2 ) + d(v 3 ) + d(v 4 ) + d(v 5 ) + d(v 6 ) = 2e Now, given 2 vertices are of degree 4 and 4 vertices are of degree 2. Hence the above equation, (4 + 4) + ( ) = 2e 16 = 2e e = 8. Hence the number of edges in a graph with 6 vertices with given condition is 8. Two such graphs are shown below in Figure (11). Fig. 11. Problem 1.4. How many vertices are needed to construct a graph with 6 edges in which each vertex is of degree 2. Solution. Suppose these are P vertices in the graph with 6 degree. Also given the degree of each vertex is 2. By handshaking lemma, P i = 1 deg ( v ) = 2q= 2 6 i d(v 1 ) + d(v 2 ) d(v n ) = = 12 2P = 12 P = 6 vertices are needed.
10 10 GRAPH THEORY WITH APPLICATIONS Problem 1.5. It is possible to construct a graph with 12 vertices such that 2 of the vertices have degree 3 and the remaining vertices have degree 4. Solution. Suppose it is possible to construct a graph with 12 vertices out of which 2 of them are having degree 3 and remaining vertices are having degree 4. Hence by handshaking lemma, 12 dv ( i ) = 2e where e is the number of edges i = 1 According to given conditions (2 3) + (10 4) = 2e = 2e 2e = 46 e = 23 It is possible to construct a graph with 23 edges and 12 vertices which satisfy given conditions. Problem 1.6. It is possible to draw a simple graph with 4 vertices and 7 edges? Justify. Solution. In a simple graph with Pvertices, the maximum number of edges will be P(P 1) Hence a simple graph with 4 vertices will have at most = 6 edges. 2 Therefore, the simple graph with 4 vertices cannot have 7 edges. Hence such a graph does not exist. Problem 1.7. Show that the maximum degree of any vertex in a simple graph with P vertices is (P 1). Solution. Let G be a simple graph with Pvertices. Consider any vertex v of G. Since the graph is simple (i.e., without self loops and parallel edges), the vertex v can be adjacent to atmost remaining (P 1) vertices. Hence the maximum degree of any vertex in a simple graph with P vertices is (P 1). Problem 1.8. Write down the vertex set and edge set of the following graphs shown in Figure 12(a) and 12(b). Fig. 12.
(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
More informationClass 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
More informationChapter 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
More informationBasic 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
More informationCSE 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!
More informationTheorem 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
More informationThe 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
More informationPlanarity 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?
More informationGraph 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.
More information1 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
More informationSolutions 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.
More informationHandout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs
MCS236: 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
More informationLong questions answer Advanced Mathematics for Computer Application If P= , find BT. 19. If B = 1 0, find 2B and 3B.
Unit1: 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
More informationV. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005
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
More informationGraph. 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
More information14. Trees and Their Properties
14. Trees and Their Properties This is the first of a few articles in which we shall study a special and important family of graphs, called trees, discuss their properties and introduce some of their applications.
More information. 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
More informationGraph 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
More informationPartitioning edgecoloured complete graphs into monochromatic cycles and paths
arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edgecoloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political
More information3. 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
More informationDiscrete 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
More informationMidterm 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
More information2.1 You need to know how graphs and networks can be used to create mathematical
fter completing this chapter you should: 1 know how graphs and networks can be used to create mathematical models 2 know some basic terminology used in graph theory 3 know some special types of graph 4
More informationGraph 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
More informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationEuler 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
More informationThe Mathematics of Origami
The Mathematics of Origami Sheri Yin June 3, 2009 1 Contents 1 Introduction 3 2 Some Basics in Abstract Algebra 4 2.1 Groups................................. 4 2.2 Ring..................................
More informationPlanar Tree Transformation: Results and Counterexample
Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem
More informationHomework 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
More information136 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
More informationA REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries
Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do
More informationCMSC 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
More informationIntroduction 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
More informationHOMEWORK #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
More informationJust 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
More informationEuclidean Geometry. We start with the idea of an axiomatic system. An axiomatic system has four parts:
Euclidean Geometry Students are often so challenged by the details of Euclidean geometry that they miss the rich structure of the subject. We give an overview of a piece of this structure below. We start
More information6.3 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
More information6. Planarity. Fig Fig. 6.2
6. Planarity Let G(V, E) be a graph with V = {v 1, v 2,..., v n } and E = {e 1, e 2,..., e m }. Let S be any surface (like the plane, sphere) and P = {p 1, p 2,..., p n } be a set of n distinct points
More informationGRAPH 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.
More informationCOLORED 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
More information6.042/18.062J 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
More informationChapter 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:
More informationdiffy 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.
More informationCSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24
CSL851: Algorithmic Graph Theory Semester I 20132014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have
More informationSimple Graphs Degrees, Isomorphism, Paths
Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week MultiGraph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1
More informationTopological Entropy of Golden Mean Lookalike Shift Spaces
International Journal of Mathematics and Statistics Invention (IJMSI) EISSN: 2321 4767 PISSN: 23214759 Volume 2 Issue 7 July 2014 PP0510 Topological Entropy of Golden Mean Lookalike Shift Spaces Khundrakpam
More informationMath 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
More information7. Colourings. 7.1 Vertex colouring
7. Colourings Colouring is one of the important branches of graph theory and has attracted the attention of almost all graph theorists, mainly because of the four colour theorem, the details of which can
More informationTotal colorings of planar graphs with small maximum degree
Total colorings of planar graphs with small maximum degree Bing Wang 1,, JianLiang Wu, SiFeng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong
More informationAxiom A.1. Lines, planes and space are sets of points. Space contains all points.
73 Appendix A.1 Basic Notions We take the terms point, line, plane, and space as undefined. We also use the concept of a set and a subset, belongs to or is an element of a set. In a formal axiomatic approach
More informationSets 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.
More informationCOMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH. 1. Introduction
COMBINATORIAL PROPERTIES OF THE HIGMANSIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the HigmanSims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact
More informationChapter 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.
More informationGraph 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 pairwise relations between
More informationGraph 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
More informationSCORE SETS IN ORIENTED GRAPHS
Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in
More informationColoring 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@unilj.si Abstract A simple characterization
More information4 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
More informationTrees 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
More informationSAMPLES OF HIGHER RATED WRITING: LAB 5
SAMPLES OF HIGHER RATED WRITING: LAB 5 1. Task Description Lab report 5. (April 14) Graph Coloring. Following the guidelines, and keeping in mind the assessment criteria, using where appropriate experimental
More informationMultiplicity. Chapter 6
Chapter 6 Multiplicity The fundamental theorem of algebra says that any polynomial of degree n 0 has exactly n roots in the complex numbers if we count with multiplicity. The zeros of a polynomial are
More informationNotes on polyhedra and 3dimensional geometry
Notes on polyhedra and 3dimensional geometry Judith Roitman / Jeremy Martin April 23, 2013 1 Polyhedra Threedimensional geometry is a very rich field; this is just a little taste of it. Our main protagonist
More informationMGF 1107 CH 15 LECTURE NOTES Denson. Section 15.1
1 Section 15.1 Consider the house plan below. This graph represents the house. Consider the mail route below. This graph represents the mail route. 2 Definitions 1. Graph a structure that describes relationships.
More informationOn planar regular graphs degree three without Hamiltonian cycles 1
On planar regular graphs degree three without Hamiltonian cycles 1 E. Grinbergs Computing Centre of Latvian State University Abstract. Necessary condition to have Hamiltonian cycle in planar graph is given.
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationGraphs 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
More informationPythagorean 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
More informationChapter I Logic and Proofs
MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than
More informationAbout the Tutorial. Audience. Prerequisites. Disclaimer & Copyright
About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a readerfriendly style, it covers the types of graphs, their properties, trees, graph traversability,
More informationPlanar 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
More informationSTRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2
STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of xaxis in anticlockwise direction, then the value of tan θ is called the slope
More informationA MEASURE OF GLOBAL EFFICIENCY IN NETWORKS. Aysun Aytac 1, Betul Atay 2. Faculty of Science Ege University 35100, Bornova, Izmir, TURKEY
International Journal of Pure and Applied Mathematics Volume 03 No. 05, 670 ISSN: 38080 (printed version); ISSN: 343395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/0.73/ijpam.v03i.5
More informationGraph 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
More information2.3 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
More informationSpecial Classes of Divisor Cordial Graphs
International Mathematical Forum, Vol. 7,, no. 35, 737749 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
More information10. 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
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationChapter 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.
More informationExtremal Wiener Index of Trees with All Degrees Odd
MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287292 ISSN 03406253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of
More informationA 2factor in which each cycle has long length in clawfree graphs
A factor in which each cycle has long length in clawfree graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science
More informationOn end degrees and infinite cycles in locally finite graphs
On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel
More informationConnectivity 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
More informationSHARP 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.,
More informationSocial 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
More informationDO NOT REDISTRIBUTE 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
More information6. 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
More informationRelations 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
More informationOn Integer Additive SetIndexers of Graphs
On Integer Additive SetIndexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A setindexer of a graph G is an injective setvalued function f : V (G) 2 X such that
More information1. Relevant standard graph theory
Color identical pairs in 4chromatic graphs Asbjørn Brændeland I argue that, given a 4chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4coloring
More information5. GRAPHS ON SURFACES
. GRPHS ON SURCS.. Graphs graph, by itself, is a combinatorial object rather than a topological one. But when we relate a graph to a surface through the process of embedding we move into the realm of topology.
More informationSequences and Series
Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will
More informationPRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT
UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,
More information16.1. Sequences and Series. Introduction. Prerequisites. Learning Outcomes. Learning Style
Sequences and Series 16.1 Introduction In this block we develop the ground work for later blocks on infinite series and on power series. We begin with simple sequences of numbers and with finite series
More informationIncenter Circumcenter
TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. The radius of incircle is
More informationNetwork/Graph Theory. What is a Network? What is network theory? Graphbased representations. Friendship Network. What makes a problem graphlike?
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 Graphbased representations Representing a problem
More informationVectors, Gradient, Divergence and Curl.
Vectors, Gradient, Divergence and Curl. 1 Introduction A vector is determined by its length and direction. They are usually denoted with letters with arrows on the top a or in bold letter a. We will use
More information7. The GaussBonnet theorem
7. The GaussBonnet theorem 7. Hyperbolic polygons In Euclidean geometry, an nsided polygon is a subset of the Euclidean plane bounded by n straight lines. Thus the edges of a Euclidean polygon are formed
More informationCOUNTING 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
More informationPrinciple of (Weak) Mathematical Induction. P(1) ( n 1)(P(n) P(n + 1)) ( n 1)(P(n))
Outline We will cover (over the next few weeks) Mathematical Induction (or Weak Induction) Strong (Mathematical) Induction Constructive Induction Structural Induction Principle of (Weak) Mathematical Induction
More informationCombinatorics: 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
More information