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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 Mária Markošová

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

3 Definition: A graph G consists of vertex set V(G) and edge set E(G) and edges are defined by vertices pairs. supgraph loop multiple edge Oriented graph: if edges have orientation. Simple graph: graph without loops and multiple edges, no edge orientation.

4 w Vertex degree a x c b d y e z Degree of a vertex x. k(x) is a number of edges incident with the vertex x. w Oriented graph k in x x b a c x y e d z In degree of vertex x, : Number of edges leading to the vertex x. k out Out degree of vertex x, : Number of edges leading out of vertex x. k x k x k x in out

5 Example. k(x)= 5 x k x k x k x in out k k in out x? x?

6 Oriented graphs: G(V,E), where V is a set of vertices, E is a set of ordered pairs e=(u,v), u, v are endpoints of edge and their order is fixed. Simple graphs: without oriented edges, no loops, no multiple edges: in what follows we shall deal with such graphs, if not given otherwise. Lemma: Unoriented graph has even number of vertices with odd degree.

7 Complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. Complete graph of N vertices is denoted as K N K 4 K K 7 2 The number of edges in the complete graph: E N 2 N N 2 What is the degree of each node in K N?

8 Regular graph is a graph where each vertex has the same number of neighbors, that means each vertex has the same degree. What about the complete graph? Is the complete graph also a regular graph?

9 Complementary graphs What is the complementary graph of? K 2 Are there a graphs, which have the same complementary graphs? Comlementary graphs mapped on each other create complete graph, in which each vertex is connected to each other vertex.

10 Bipartite graph Graphs, in which V(G) (vertex set) is the union of the two disjoint independent sets (no edges between nodes in the set). people jobs What type of the graph is also a bipartite graph? Find an example. men women authors papers Find another examples at home.

11 Tree graph: Undirected graph in which two vertices are connected exactly by one simple path. Is the tree on fig a bipartite graph? Every tree is a bipartite graph.

12 Spanning tree: Spanning tree of the connected undirected graph G is a tree composed of all vertices and some edges of the graph G. A spanning tree of a connected graph G can also be defined as a maximal set of edges of G that contains no cycle, or as a minimal set of edges that connect all vertices.

13 Graph representation Loopless graph: Graph without loops, multiple edges are allowed w x y z a b c d e 2 2 z y x w W x z y A(G) adjacency matrixnumber of edges between defined endpoints z y x w e d c b a M(G) incidence matrix- One, if vertex is the endpoint of an edge, zero if not

14 w x y z a b c d e 2 2 z y x w W x z y A(G)= Every adjacency matrix is symetric (no oriented graphs). The degree of vertex x is a sum of entries of row x in both A(G) and M(G). z y x w e d c b a M(G)= 3 2 x k

15 Isomorphic graphs Two graphs G and H are isomorphic, if we are able to map one to the another (e.g. to find a bijection f: V(G) V(H), and if uv is from E(G) then f(u)f(v) is from E(H)) G H Adjacency matrix of G is the same as of H if we reorder the vertices

16 Self complementary graphs Definition: A graph is self complementary if it is isomorphic to its complement. 5 a b 2 c 4 e 3 d

17 Isomorfism has.reflexive, 2.symetric and 3. transitive property:. Each graph is isomorphic to itself. 2. If f : V G V H is an isomorphism from G to H, then f is an isomorphism from H to G. f : V F V G g V G V H 3. Suppose that, and : are isomorphisms then composition f g is isomorphism from F to G. Isomorphic graphs have isomorphic complements. Sometimes it is easier to test the complements to decide, whether two graphs are isomorfic or not.

18 Are they isomorphic? 6 a b 5 2 e f 4 3 d c 5 6 a b Complementary 2 graphs e f 4 3 d c

19 To prove, that graphs are NOT isomorphic is enough to find, that in some structural properties ( e.g. number of edges, supgraphs, complements etc.) they differ.

20 Isomorphic classes P C n K K n n r, s - path with n vertices - cycle with n vertices - complete graph with n vertices - complete bipartite graph, r,s are indexes of two sets of vertices CK P 5 K 2,3

21 What is the number of simple graphs which can be created on a set of N vertices? m 2 N 2 From a set of four vertices we can create 64 different simple graphs, belonging to isomorphism classes. Here are the representatives of all classes

22 Connected and disconnected graphs Definition: A graph G is connected if it has u,v path whenever u, v are from V(G). Othervise G is disconnected. e e2 x e6 w e5 y e4 e7 e3 z

23 v, e, v,..., e, v k k Definition: A walk is a list of vertices and edges such that for i k the edge has endpoints v, i and v i. Edges and vertices can be repeated. e i A u,v-walk or u,v trail has first vertex u and last vertex v. A trail is a walk with no repeated edge. A u,v- path is a path whose vertices of degree one are u,v (endpoints). Other vertices are internal. Edges and vertices are not repeated. A walk and trial is closed if its endpoints are the same.

24 Euler s question Konigsberg bridges Walk of lenght 4: x.e2,w,e5,y,e6,x,e2,w Closed walk of lenght 5: x x.e2,w,e5,y,e6,x,e,w,e2,x e e2 e6 Trail of length 4 w e5 walk cycle trail y x.e2,w,e5,y,e6,x,e,w e3 e4 z e7 Cycle of lenght 3: Subgraph consisting of edges e,e6,e5 and vertices w,x,y. Deleting one of its edges we get a path of length 2.

25 Eulerian trial: Trial which visits every edge at most once. Euler s lemma: Connected graph has closed eulerian trial if and only if all of its vertices have even degree. x y z w e e 2 e 3 e 4 e 5 e 6 e 7 x y z w e e 3 e 6 e 7 Why?

26 Connectivity and adjacency G has w,z- path xw E G w and z are connected w is connected to z x and w are adjacent x is joined to w x is adjacent to w w a x c b d y e z

27 Component of graph Definitions: Maximal connected subgraph of graph G is a subgraph, which is connected and is not contained in any other connected subgraph of G. The components of graph G are its maximal connected subgraphs. 4 components of graph G

28 Proposition: Every graph with N nodes and m edges has at least N-m components. Proof: N isolated vertices create a graph with N components. Each added edge links vertices in the same component, or in different components merging them in one. So adding an edge decreases the number of components by or one. If we add m edges, the number of components is at least N-m. Definition: Cut edge or cut vertex is an edge or vertex whose deletion increases the number of components.

29 Cut vertex increases the number of subgraphs by many Cut edge increases the number of subgraphs always by one

30 Vertex degrees Degree of vertices are fundamental parameters of graph. k G v G G Definitions: Degree of a vertex v of graph G Maximum degree in graph G Minimum degree in graph G Graph G is regular if all degrees are equal. Graph G is k-regular if the common degree is k. N(v) neigborhood of vertex v, set of vertices adjacent to v N(G)- order of G, number of vertices in G e(g)- size of G, number of edges in G

31 First teorem of graph theory If G is a graph, then : Proof: each edge adds to the degree of two vertices. What can be concluded from the theorem?. Average vertex degree is: 2. Every graph has an even number of vertices of odd degree. No graph of odd order k is regular with the odd degree. 3. A k regular graph of with N vertices has Nk/2 edges. G e v k G V v 2 G G N G e G, G N G e 2 2

32 Proposition: If k is greather then zero, then a k regular bipartite graph has the same number of vertices in each partite set. Proof : Let G be an X,Y bigraph. Counting the edges according to their endpoints in X yealds e G k X.Counting them by their endpoints in Y yealds e G k Y. Thus k X k Y X Y, k And therefore. X Y

33 Extremal problems Extremal problem: asks for maximal or minimal value of a function over a class of objects.. Maximal number of edges in a simple graph (having no loops and multiple edges )is N N! N. N 2 N 2! 2! 2 2. The minimum number of edges in a connected graph with N vertices is N-. 3. If G is a simple N vertex graph with, then G is connected. G N 2

34 Model A: Let us have a set of N vertices and the probability p(n)=p, that each pair of vertices is connected by an edge. Each graph with m edges has a probability of occurrence p m N m p 2 The random variable G p denotes a realization of such graph. m=4

35 Model B: Let us have N vertices and m=m(n) edges and each such graph occurs with probability, where n is a possible number of edges if each vertex is connected to each and. Random variable denotes a graph generated in this way. Both models are the same for large number of vertices and probability m n 2 N n m G 2 / / N m n m p

36 Random graph theory studies the properties of the probability space associated with graphs with N nodes,. N Eordos, Renyi: Almost every graph has a property Q if the probability of having Q approaches one, as. N What is Q? Examples: -is a typical graph connected -does typical graph contain triangles, trees, etc. -etc.

37 Evolution of graph: One starts with isolated points and edges are succesively added. This corresponds to groving probability p in the A model. Goal: At what probability p particullar property Q in a graph arises? Many important properties appear suddenly, at certain critical probability p c N. lim N P N, p Q Probability to have graph with quality Q, number of nodes N and edge probability between nodes p, if, if p p c p p c N N N N

38 It can be shown, that the number of subgraphs F contained in graph G having number of nodes N and probability p of two vertices being connected is: E n F N a p m a m That is: The number of subgraph is finite if -number of nodes in subgraph -number of isomorfic subgraphs -number of edges p cn m

39 Evolutionary stage of graphs: It is a range of values for m(n) or p(n) in which a structural description of a typical graph does not change too much.. If there is a huge amount of vertices compared to the number of edges, then adding an edge to a randomly chosen pair makes an isolated edge. 2. Then, for certain threshold probabilities (or threshold number of edges) subtrees with vertices appear. Graph has no cycles. p c / N cn Number of nodes in tree Number of edges in tree

40 N c N p c / c 3. For probability with cycles start to appear. 4. The critical probability of having a complete subgraph of order is p c 2 / N cn cycle

41 Evolution of graphs intuitively

42 Conclusion: What to remember: Types of graphs (unoriented, oriented, simple, complete, bipartite... Important properties (degree, connectivity... ) What you need to understand: Lemmas at least on the intuitive level

### 1 Digraphs. Definition 1

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

### 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

### 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

### (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

### 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

### 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

### 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

### Chapter 2. Basic Terminology and Preliminaries

Chapter 2 Basic Terminology and Preliminaries 6 Chapter 2. Basic Terminology and Preliminaries 7 2.1 Introduction This chapter is intended to provide all the fundamental terminology and notations which

### 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

### 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

### 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).

### 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.

### 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

### 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

### 1. Relevant standard graph theory

Color identical pairs in 4-chromatic graphs Asbjørn Brændeland I argue that, given a 4-chromatic graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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?

### 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

### 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

### 1 Plane and Planar Graphs. Definition 1 A graph G(V,E) is called plane if

Plane and Planar Graphs Definition A graph G(V,E) is called plane if V is a set of points in the plane; E is a set of curves in the plane such that. every curve contains at most two vertices and these

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

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

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

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

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

### 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

### 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

### 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

### 8. Matchings and Factors

8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

### 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

### 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

### 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

### 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

### 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

### Some Results on 2-Lifts of Graphs

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

### 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

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### MGF 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.

### 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

### 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.

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

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

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

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

### 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!

### On one-factorizations of replacement products

Filomat 27:1 (2013), 57 63 DOI 10.2298/FIL1301057A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On one-factorizations of replacement

### 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

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

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

### 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

### 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

### 6. 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

### Graph Theory. Euler tours and Chinese postmen. John Quinn. Week 5

Graph Theory Euler tours and Chinese postmen John Quinn Week 5 Recap: connectivity Connectivity and edge-connectivity of a graph Blocks Kruskal s algorithm Königsberg, Prussia The Seven Bridges of Königsberg

### 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

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

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

### 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

### 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

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

### 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

### 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

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

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

### 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:

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter

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

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

### Zero-Sum Magic Labelings and Null Sets of Regular Graphs

Zero-Sum Magic Labelings and Null Sets of Regular Graphs Saieed Akbari a,c Farhad Rahmati b Sanaz Zare b,c a Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran b Department

### Finding and counting given length cycles

Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

### 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

### Planar 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

### 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

### 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

About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a reader-friendly style, it covers the types of graphs, their properties, trees, graph traversability,

### 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

### Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

### On a tree graph dened by a set of cycles

Discrete Mathematics 271 (2003) 303 310 www.elsevier.com/locate/disc Note On a tree graph dened by a set of cycles Xueliang Li a,vctor Neumann-Lara b, Eduardo Rivera-Campo c;1 a Center for Combinatorics,

### Coloring Eulerian triangulations of the projective plane

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

### A tree can be defined in a variety of ways as is shown in the following theorem: 2. There exists a unique path between every two vertices of G.

7 Basic Properties 24 TREES 7 Basic Properties Definition 7.1: A connected graph G is called a tree if the removal of any of its edges makes G disconnected. A tree can be defined in a variety of ways as

### 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

### Diameter and Treewidth in Minor-Closed Graph Families, Revisited

Algorithmica manuscript No. (will be inserted by the editor) Diameter and Treewidth in Minor-Closed Graph Families, Revisited Erik D. Demaine, MohammadTaghi Hajiaghayi MIT Computer Science and Artificial

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### On Total Domination in Graphs

University of Houston - Downtown Senior Project - Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper

### 2.1 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

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, steen@cs.vu.nl Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction

### 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,

### Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

- 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

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

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

### A threshold for the Maker-Breaker clique game

A threshold for the Maker-Breaker clique game Tobias Müller Miloš Stojaković October 7, 01 Abstract We study the Maker-Breaker k-clique game played on the edge set of the random graph G(n, p. In this game,

### 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

### 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

### 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

### Linear Algebra and its Applications

Linear Algebra and its Applications 438 2013) 1393 1397 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Note on the

### 9.1. Vertex colouring. 9. Graph colouring. Vertex colouring.

Vertex colouring. 9. Graph colouring k-vertex-critical graphs Approximation algorithms Upper bounds for the vertex chromatic number Brooks Theorem and Hajós Conjecture Chromatic Polynomials Colouring of

### 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

### Spectral graph theory

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

### Unicyclic Graphs with Given Number of Cut Vertices and the Maximal Merrifield - Simmons Index

Filomat 28:3 (2014), 451 461 DOI 10.2298/FIL1403451H Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Unicyclic Graphs with Given Number

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA