Graph. Notation: E = * *1, 2+, *1, 3+, *2, 3+, *3, 4+ +


 Stanley Long
 11 months ago
 Views:
Transcription
1 Graph graph is a pair (V, E) of two sets where V = set of elements called vertices (singl. vertex) E = set of pairs of vertices (elements of V) called edges Example: G = (V, E) where V = *,,, + E = * *, +, *, +, *, +, *, + + Notation: we write G = (V, E) for a graph with vertex set V and edge set E V(G) is the vertex set of G, and E(G) is the edge set of G drawing of G
2 Types of graphs Undirected graph E = **,+, *,+, *,+, *,++ SYMMETRIC & IRREFLEXIVE E(G) = set of unordered pairs Pseudograph (allows loops) loop = edge between vertex and itself Directed graph E(G) = set of ordered pairs Multigraph (IRREFLEXIVE) RELATION E(G) is a multiset E = **,+, *,+, *,+, E = **,+, *,+, *,+, *,+,,, *,+, *,+,,, *,+, *,+, *,+, *,++ *,++ 0 SYMMETRIC? E = *(,), (,), (,), (,), (,)+
3 More graph terminology simple graph (undirected, no loops, no parallel edges) for edge *u, v+ E(G) we say: u and v are adjacent u and v are neighbours u and v are endpoints of *u, v+ we write uv E G for simplicity N(v) = set of neighbours of v in G deg (v) = degree of v is the number of its neighbours, ie. N(v) is adjacent to and N() = *,+ N() = *,+ N() = *,,+ deg () = N() = *+ deg = deg () = deg () = = f() = f() = f() = G = (V, E ) is isomorphic to G = (V, E ) if there exists a bijective mapping f: V V such that uv E(G ) if and only if f u f(v) E(G )
4 Isomorphism 6 6 (,,,,,) degree sequence (,,,,,) (,,,,,) 6 if f is an isomorphism deg (u) = deg (f(u)) i.e., isomorphic graphs have same degree sequences only necessary not sufficient!!! (,,,,,,,) (,,,,,,,)
5 Isomorphism How many pairwise nonisomorphic graphs on n vertices are there? nonisomorphic graphs (without labels) all graphs (labeled) the complement of G = (V, E) is the graph G = (V, E) where E = u, v *u, v+ E+ E(G) = **,+, *,+, *,+, *,++ E(G ) = **,+, *,+, *,+, *,+, *,+, *,++ G G
6 Isomorphism How many pairwise nonisomorphic graphs on n vertices are there? selfcomplementary graph
7 Isomorphism Are there self complementary graphs on vertices?,,,... 6 vertices? No, because in a selfcomplementary graph G = (V, E) E = E and E + E = V 6 but = is odd 7 No, since =... 7 vertices?... 8 vertices? Yes, there are 0, 6,, 7, 6 8, 7, Yes,,,,
8 Handshake lemma Lemma. Let G = (V, E) be a graph with m edges. deg v = m v V Proof. Every edge *u, v+ connects vertices and contributes to exactly to deg (u) and exactly to deg (v). In other words, in v V deg (v) every edge is counted twice. How many edges has a graph with degree sequence:  (,,,,,)?  (,,,,)?  (0,,,)? Corollary. In any graph, the number of vertices of odd degree is even. Corollary. Every graph with at least vertices has vertices of the same degree. deg () = deg () = deg () = deg = m = 6
9 Handshake lemma Is it possible for a selfcomplementary graph with 00 vertices to have exactly one vertex of degree 0? Answer. No f isomorphism between G and G v is a unique vertex of degree 9 in G u the degree seq. of G and G (,,, 0,9,,, ) v G N(u) 0 V(G)\N(u) v, f(v) = u 9 we conclude f(u) = v N G (u) uv E(G) f(u)f(v) E(G ) but f(u)f(v) = vu E(G ) uv E(G) definition of isomorphism u G definition of complement 9
10 Subgraphs Let G = (V, E) be a graph a subgraph of G is a graph G = (V, E ) where V V and E E an induced subgraph (a subgraph induced on A V) is a graph G,A = (A, E A ) where E A = E A A subgraph induced subgraph on A = *,,+
11 Walks, Paths and Cycles Let G = (V, E) be a graph a walk (of length k) in G is a sequence v, e, v, e,, v k, e k, v k where v,, v k V and e i = *v i, v i+ + E for all i *.. k + a path in G is a walk where v,, v k are distinct (does not go through the same vertex twice) a closed walk in G is a walk with v = v k (starts and ends in the same vertex) a cycle in G is a closed walk where k and v,, v k are distinct,{,},,{,},,{,}, is a walk (not path),{,},,{,},,{,}, is a cycle
12 Special graphs P n path on n vertices C n cycle on n vertices K n complete graph on n vertices B n hypercube of dimension n V B n = 0, n (a,, a n ) adjacent to b,, b n if a i s and b i s differ in exactly once n i= a i b i = (0,,0,0) adjacent to (0,,0,) not adjacent to (0,,,) P C K B
13 Connectivity a graph G is connected if for any two vertices u, v of G there exists a path (walk) in G starting in u and ending in v,{,},,{,}, path from to connected components no path from to connected not connected = disconnected a connected component of G is a maximal connected subgraph of G
14 Connectivity Show that the complement of a disconnected graph is connected! u u u v v w w G G G What is the maximum number of edges in a disconnected graph? k n k max k k + n k = n maximum when k = What is the minimum number of edges in a connected graph? path P n on n vertices has n edges cannot be less, why? keep removing edges so long as you keep the graph connected a (spanning) tree which has n edges
15 Distance, Diameter and Radius recall walk (path) is a sequence of vertices and edges v, e, v, e,, v k, e k, v k if edges are clear from context we write v, v,, v k length of a walk (path) is the number of edges distance d(u, v) from a vertex u to a vertex v is the length of a shortest path in G between u and v if no path exists define d u, v = d(u, v) d(,) = d(,) = matrix of distances in G distance matrix
16 Distance, Diameter and Radius eccentricity of a vertex u is the largest distance between u and any other vertex of G; we write ecc u = max d(u, v) v diameter of G is the largest distance between two vertices diam G = max d u, v = max ecc(v) u,v v radius of G is the minimum eccentricity of a vertex of G rad(g) = (as witnessed by called a centre) rad G = min v ecc(v) diam(g) = (as witnessed by, called a diametrical pair) Find radius and diameter of graphs P n, C n, K n, B n Prove that rad(g) diam(g) rad(g) ecc() = ecc() = ecc() = ecc() =
17 Independent set and Clique clique = set of pairwise adjacent vertices independent set = set of pairwise nonadjacent vertices clique number ω(g) = size of largest clique in G independence number α(g) = size of largest independent set in G *,,+ is a clique α G = ω G = *,+ is an independent set Find α(p n ), α(c n ), α K n, α(b n ) ω(p n ), ω(c n ), ω(k n ), ω(b n ) P
18 Bipartite graph a graph is bipartite if its vertex set V can be partitioned into two independent sets V, V ; i.e., V V = V bipartite not bipartite V V V V but now V is not an independent set because *,+ E(G) V
19 Bipartite graph a graph is bipartite if its vertex set V can be partitioned into two independent sets V, V ; i.e., V V = V K, K, K, Which of these graphs are bipartite: P n, C n, K n, B n? K n,m = complete bipartite graph V K n,m = *a a n, b b m + E K n,m = a i b j i.. n, j.. m + K,
20 Bipartite graph Theorem. A graph is bipartite it has no cycle of odd length. Proof. Let G = (V, E). We may assume that G is connected. any cycle in a bipartite graph is of even length 6 k odd k Assume G has no oddlength cycle. Fix a vertex u and put it in V. Then repeat as long as possible: u  take any vertex in V and put its neighbours in V, or  take any vertex in V and put its neighbours in V. Afterwards V V = V because G is connected. If one of V or V is not an independent set, then we find in G an oddlength closed walk cycle, impossible.? So, G is bipartite (as certified by the partition V V = V)
Basic Combinatorics. Math 40210, Section 01 Fall Basic graph definitions
Basic Combinatorics Math 40210, Section 01 Fall 2012 Basic graph definitions It may seem as though the beginning of graph theory comes with a lot of definitions. This may be so, but hopefully by repeatedly
More informationCHAPTER 2. Graphs. 1. Introduction to Graphs and Graph Isomorphism
CHAPTER 2 Graphs 1. Introduction to Graphs and Graph Isomorphism 1.1. The Graph Menagerie. Definition 1.1.1. A simple graph G = (V, E) consists of a set V of vertices and a set E of edges, represented
More informationGraph. Graph Theory. Adjacent, Nonadjacent, Incident. Degree of Graph
Graph Graph Theory Peter Lo A Graph (or undirected graph) G consists of a set V of vertices (or nodes) and a set E of edges (or arcs) such that each edge e E is associated with an unordered pair of vertices.
More informationGraph Theory. Clemens Heuberger and Stephan Wagner. AIMS, January/February 2009
Graph Theory Clemens Heuberger and Stephan Wagner AIMS, January/February 2009 1 Basic Definitions Definition 1.1 A graph is a pair G = (V (G),E(G)) of a set of vertices V (G) and a set of edges E(G), where
More informationHonours Graph Theory
University of Zimbabwe HMTH215 Graph Theory Honours Graph Theory Author: P. Mafuta Department: Mathematics April 6, 2016 Chapter 1 Introduction: Basic Graph Theory This course serves to answer many questions
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 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 informationDiscrete Mathematics (Math 510) Fall Definitions and Theorems
Discrete Mathematics (Math 510) Fall 2014 Definitions and Theorems Gerald Hoehn October 12, 2014 Chapter 1 Graph Theory 1.1 Basics Definition 1.1. A graph G is an ordered pair (V,E) of disjoint finite
More information3. Representing Graphs and Graph Isomorphism. Discussion
3. REPRESENTING GRAPHS AND GRAPH ISOMORPHISM 195 3. Representing Graphs and Graph Isomorphism 3.1. Adjacency Matrix. Definition 3.1.1. The adjacency matrix, A = [a ij ], for a simple graph G = (V, E),
More information1 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
More informationMATH 2420 Discrete Mathematics Lecture notes
MATH 2420 Discrete Mathematics Lecture notes Graphs Objectives Graphs 1. Identify loops, parallel edges, etc. in a graph. 2. Draw the complete graph on n vertices, and the complete bipartite graph on (m,n)
More information2. Graph Terminology
2. GRAPH TERMINOLOGY 186 2. Graph Terminology 2.1. Undirected Graphs. Definitions 2.1.1. Suppose G = (V, E) is an undirected graph. (1) Two vertices u, v V are adjacent or neighbors if there is an edge
More information1 Connected simple graphs on four vertices
1 Connected simple graphs on four vertices Here we briefly answer Exercise 3.3 of the previous notes. Please come to office hours if you have any questions about this proof. Theorem 1.1. There are exactly
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 informationIn example: V(G) = {a,b,c,d,e,f,g}, E(G) = {u,v,w,x,y}, ψ G maps a uv (i.e. ψ G (a) = {u,v} = uv), b vw, c vx, d wy, e wy, f xy.
BASIC GRAPH THEORY DEFINITIONS If book and instructor disagree, follow instructor! Graphs graph G consists of vertex set V(G), edge set E(G), incidence relation ψ G mapping each edge to unordered pair
More informationGRAPH THEORY STUDY GUIDE
GRAPH THEORY STUDY GUIDE 1. Definitions Definition 1 (Partition of A). A set A = A 1,..., A k of disjoint subsets of a set A is a partition of A if A of all the sets A i A and A i for every i. Definition
More informationOutline 1.1 Graphs and Digraphs 1.2 Common Families of Graphs 1.4 Walks and Distance 1.5 Paths, Cycles, and Trees
GRAPH THEORY LECTURE 1 INTRODUCTION TO GRAPH MODELS Abstract. Chapter 1 introduces some basic terminology. 1.1 is concerned with the existence and construction of a graph with a given degree sequence.
More informationGRAPH THEORY: INTRODUCTION
GRAPH THEORY: INTRODUCTION DEFINITION 1: A graph G consists of two finite sets: a set V (G) of vertices a set E(G) of edges, where each edge is associated with a set consisting of either one or two vertices
More informationGraph Theory. 1 Defining and representing graphs
Graph Theory 1 Defining and representing graphs A graph is an ordered pair G = (V, E), where V is a finite, nonempty set of objects called vertices, and E is a (possibly empty) set of unordered pairs
More informationSubgraphs and Paths and Cycles
15 October, 2015 Subgraphs Definition Let G = (V, E) be a graph. Graph H = (V, E ) is a subgraph of G if V V and E E. Subgraphs Definition Let G = (V, E) be a graph. Graph H = (V, E ) is a subgraph of
More informationCS 441 Discrete Mathematics for CS Lecture 25. Graphs. CS 441 Discrete mathematics for CS. Definition of a graph
CS 441 Discrete Mathematics for CS Lecture 25 Graphs Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Definition of a graph Definition: A graph G = (V, E) consists of a nonempty set V of vertices
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. 1.1 Graph: Prepared by M. Aurangzeb Updated: Monday, March 15, 2010
Prepared by M. Aurangzeb Updated: Monday, March 15, 2010 Graph Theory Graph theory is the branch of mathematics which deals with entities and their mutual relationships. The entities are represented by
More informationVertex colouring and chromatic polynomials
Vertex colouring and chromatic polynomials for Master Course MSM380 Dong Fengming National Institute of Education Nanyang Technological University Contents 1 Vertex Colourings 1 1.1 Definition...............................
More information1 Cliques and Independent Sets. Definition 1
1 Cliques and Independent Sets Definition 1 A set of vertices is called independent if no two vertices in the set are adjacent. A set of vertices is called a clique if every two vertices in the set are
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 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 informationIn this section, we shall assume (except where noted) that graphs are loopless.
6 Graph Colouring In this section, we shall assume (except where noted) that graphs are loopless. Upper and Lower Bounds Colouring: A kcolouring of a graph G is a map φ : V (G) S where S = k with the
More information(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 information1.5 Problems and solutions
15 PROBLEMS AND SOLUTIONS 11 15 Problems and solutions Homework 1: 01 Show by induction that n 1 + 2 2 + + n 2 = n(n + 1)(2n + 1) 01 Show by induction that n 1 + 2 2 + + n 2 = n(n + 1)(2n + 1) We ve already
More informationOn 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
More information2tone Colorings in Graph Products
tone Colorings in Graph Products Jennifer Loe, Danielle Middlebrooks, Ashley Morris August 13, 013 Abstract A variation of graph coloring known as a ttone kcoloring assigns a set of t colors to each
More information2. Determine each of the 11 nonisomorphic graphs of order 4, and give a planar representation of each.
Chapter 11 Homework. Determine each of the 11 nonisomorphic graphs of order 4, and give a planar representation of each. 3. Does there exist a graph of order 5 whose degree sequence equals (4, 4, 3,, )?
More informationThe number of edges in a bipartite graph of given radius
The number of edges in a bipartite graph of given radius P. Dankelmann, Henda C. Swart, P. van den Berg University of KwaZuluNatal, Durban, South Africa Abstract Vizing established an upper bound on the
More informationGraph 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
More informationGraph 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
More informationSquare Root Finding In Graphs
Square Root Finding In Graphs Majid Karimi, Master of Science Mathematics and Statistics Submitted in partial fulfilment of the requirements for the degree of Master of Science Faculty of Mathematics and
More information{good walks whose first step is up}
Problem 1. (20 points) Compute the number of (shortest) grid walks from (0, 0) to (9, 9) which: a) do not go through any of the other diagonal points (1, 1), (2,,..., (8, 8). Solution: If the first step
More informationGraph Theory and Metric
Chapter 2 Graph Theory and Metric Dimension 2.1 Science and Engineering Multi discipline teams and multi discipline areas are words that now a days seem to be important in the scientific research. Indeed
More informationBipartite Graphs and Problem Solving
Bipartite Graphs and Problem Solving Jimmy Salvatore University of Chicago August 8, 2007 Abstract This paper will begin with a brief introduction to the theory of graphs and will focus primarily on the
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 informationV. Adamchik 1. Graph Theory. Victor Adamchik. Fall of There are different ways to draw the same graph. Consiser the following two graphs.
V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Graph Isomorphism 2. Graph Enumeration 3. Planar Graphs Graphs Isomorphism There are different ways to draw the same graph. Consiser the
More informationKnight s Tour Problem and its Graph Analysis. Deepak Kumar Kent State University
Knight s Tour Problem and its Graph Analysis Deepak Kumar Kent State University In this Presentation: Knight s Tour and its Problem discussion Building Knight s Graph Analysis on Knight Graph Special Properties
More informationThis is an on going project. Use at your own risk. There are bound to be typos. Corrections are appreciated, and rewarded with small amounts of extra
Graph Theory Notes This is an on going project. Use at your own risk. There are bound to be typos. Corrections are appreciated, and rewarded with small amounts of extra credit, especially when they indicate
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 informationMathematics 1. Part I: Graph Theory. Exercises and problems
Bachelor Degree in Informatics Engineering Barcelona School of Informatics Mathematics 1 Part I: Graph Theory Exercises and problems February 2015 Departament de Matemàtica Aplicada 2 Universitat Politècnica
More informationMIDTERM MATH 38, SPRING 2012 SOLUTIONS
MIDTERM MATH 38, SPRING 2012 SOLUTIONS 1. [20 points] (a) Prove that if G is a simple graph of order n such that (G) + δ(g) n 1, then G is connected. (Hint: Consider a vertex of maximum degree.) (b) Show
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 informationDefinition. 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
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 informationIndependent Monopoly Size In Graphs
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1939466 Vol. 10, Issue (December 015), pp. 738 749 Applications and Applied Mathematics: An International Journal (AAM) Independent Monopoly Size
More informationMTH 548 Graph Theory Fall 2003
MTH 548 Graph Theory Fall 2003 Lesson 6  Planar graphs Planar graph plane drawing  plane graph face length of a face infinite face Thm 12: Euler. Let G be a plane drawing of a connected planar graph.
More information1.7. Isomorphic Graphs
1.7. Isomorphic Graphs Example: Consider the following graphs, are they the isomorphic, i.e. the same? No. The lefthand graph has 5 edges; the right hand graph has 6 edges. WUCT121 Graphs 25 Firstly,
More informationBOUNDARY EDGE DOMINATION IN GRAPHS
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 04874, ISSN (o) 04955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 19704 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
More informationOrdering without forbidden patterns
Ordering without forbidden patterns Pavol Hell Bojan Mohar Arash Rafiey Abstract Let F be a set of ordered patterns, i.e., graphs whose vertices are linearly ordered. An Ffree ordering of the vertices
More informationElectrical Resistance Networks and Random Walks
Electrical Resistance Networks and Random Walks Natalie Frank June 8, 205 Introduction Any networklike structure can be thought of as a graph. Thus, graphs have a wide variety of applications.a random
More informationSection 7.4 Connectivity. We extent the notion of a path to undirected graphs. An informal definition (see the text for a formal definition):
Section 7.4 Connectivity We extent the notion of a path to undirected graphs. An informal definition (see the text for a formal definition): There is a path v0, v1, v2,..., vn from vertex v0 to vertex
More informationDefinitions and examples
Chapter 1 Definitions and examples I hate definitions! Benjamin Disraeli In this chapter, we lay the foundations for a proper study of graph theory. Section 1.1 formalizes some of the graph ideas in the
More informationMAP363 Combinatorics Answers 1
MAP6 Combinatorics Answers Guidance on notation: graphs may have multiple edges, but may not have loops. A graph is simple if it has no multiple edges.. (a) Show that if two graphs have the same degree
More information1 Introduction to Graph Theory
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.
More informationGraph Colouring and Optimisation Problems
Graph Colouring and Optimisation Problems Cycles Girth Bipartite graphs Weighting Lecturer: Georgy Gimel farb COMPSCI 220 Algorithms and Data Structures 1 / 45 1 Cycles and girth 2 Bipartite graphs 3 Maximum
More informationDefinition A tree is a connected, acyclic graph. Alternately, a tree is a graph in which any two vertices are connected by exactly one simple path.
11.5 Trees Examples Definitions Definition A tree is a connected, acyclic graph. Alternately, a tree is a graph in which any two vertices are connected by exactly one simple path. Definition In an undirected
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 information(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 37 (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,..., c}.] Definition: A proper
More informationINEQUALITIES OF NORDHAUS GADDUM TYPE FOR CONNECTED DOMINATION
INEQUALITIES OF NORDHAUS GADDUM TYPE FOR CONNECTED DOMINATION H. KARAMI DEPARTMENT OF MATHEMATICS SHARIF UNIVERSITY OF TECHNOLOGY P.O. BOX 11365915 TEHRAN, I.R. IRAN S.M. SHEIKHOLESLAMI DEPARTMENT OF
More informationGlossary of graph theory From Wikipedia, the free encyclopedia
Glossary of graph theory From Wikipedia, the free encyclopedia Graph theory is a growing area in mathematical research, and has a large specialized vocabulary. Some authors use the same word with different
More informationMULTIPLE CHOICE QUESTIONS. 1) Let A and B be any two arbitrary events then which one of the following is true?
DISCRETE SRUCTURES MULTIPLE CHOICE QUESTIONS 1) Let A and B be any two arbitrary events then which one of the following is true? a. P( A intersection B) = P(A). P(B) b. P(A union B) = P(A) + P(B) c. P(AB)
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 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 4 SOLUTIONS. We then drop the leading 1 in the code and put 2 at the back of the code:
HOMEWORK 4 SOLUTIONS (1) Draw the tree whose Prüfer code is (1, 1, 1, 1, 6, 5) Solution: The given Prüfer code has six entries, therefore the corresponding tree will have 6 + = 8 entries The first number
More informationSome More Results on Facto Graphs
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 50, 24832492 Some More Results on Facto Graphs E. Giftin Vedha Merly Scott Christian College (Autonomous), Nagercoil Kanyakumari District, Tamil Nadu,
More informationConnections Between the Matching and Chromatic Polynomials
Connections Between the Matching and Chromatic Polynomials E.J. Farrell Earl Glen Whitehead JR Presented by Doron Tiferet Outline The following presentation will focus on two main results: 1. The connection
More informationSolutions to Exercises Chapter 11: Graphs
Solutions to Exercises Chapter 11: Graphs 1 There are 34 nonisomorphic graphs on 5 vertices (compare Exercise 6 of Chapter 2). How many of these are (a) connected, (b) forests, (c) trees, (d) Eulerian,
More informationOutline 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
More informationModule 3. Trees. Contents
Module Trees Contents. Definitions and characterizations............... Number of trees (Optional)................. 5 Cayley s formula........................ Kirchoffmatrixtree theorem.................
More informationThe University of Sydney MATH2009
The University of Sydney MATH2009 GRAPH THEORY Tutorial Solutions 200 1. Find a solution to the hinese Postman Problem in this graph, given that every edge has equal weight. The problem is to find the
More informationMATH 22 REVIEW. Lecture W: 11/20/2003
MATH 22 Lecture W: 11/20/2003 REVIEW Nature fits all her children with something to do; He who would write and can t write, can surely review. James Russell Lowell, A Fable for Critics Copyright 2003 Larry
More informationWeek 9: Planar and nonplanar graphs. 2 and 4, November, 2016
(1/22) MA204/MA284 : Discrete Mathematics Week 9: Planar and nonplanar graphs http://www.maths.nuigalway.ie/~niall/ma284/ 2 and 4, November, 2016 1 Definitions (again) 6. Subgraphs Named graphs 2 Planar
More informationTHE 0/1BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER
THE 0/1BORSUK 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
More informationRandom Graphs : Markov s Inequality
Graphs and Networs Lecture 11 Random Graphs : Marov s Inequality Lecturer: Daniel A. Spielman October 11, 2007 11.1 Introduction In this lecture, we will consider the ErdosRenyi model of random graphs.
More informationProblem Set #11 solutions
5.1.8. Find three nonisomorphic graphs with the same degree sequence (1, 1, 1,,, 3). There are several such graphs: three are shown below. 5.1.10. Show that two projections of the Petersen graph are isomorphic.
More informationGraph Theory Final Exam
Graph Theory Final Exam May 2, 2004 Directions. Solve the five problems below. Ask questions whenever it is not clear what is being asked of you. Each problem is worth 20 points. Notation. For a positive
More informationChapter 6 GRAPH COLORING
Chapter 6 GRAPH COLORING A kcoloring of (the vertex set of) a graph G is a function c : V (G) {1,,..., k} such that c (u) c (v) whenever u is adjacent to v. Ifakcoloring of G exists, then G is called
More information1 Cliques and Independent Sets. Definition 1
1 Cliques and Independent Sets Definition 1 A set of vertices is called independent if no two vertices in the set are adjacent. A set of vertices is called a clique if every two vertices in the set are
More informationOn size, radius and minimum degree
Discrete Mathematics and Theoretical Computer Science DMTCS vol. 16:1, 2014, 1 6 On size, radius and minimum degree Simon Mukwembi University of KwaZuluNatal, South Africa. received 9 th Feb. 2012, accepted
More informationGraphs. Discrete Mathematics (MA 2333) Faculty of Science Telkom Institute of Technology Bandung  Indonesia
Graphs Discrete Mathematics (MA 2333) Faculty of Science Telkom Institute of Technology Bandung  Indonesia Introduction Graph theory is an old subject with many modern applications. Its basic idea were
More informationVERTEXTOEDGE CENTERS W.R.T. DDISTANCE
italian journal of pure and applied mathematics n. 35 2015 (101 108) 101 VERTEXTOEDGE CENTERS W.R.T. DDISTANCE D. Reddy Babu Department of Mathematics K.L. University Vaddeswaram Guntur 522 502 India
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 informationON CHARACTERISTIC SUBGRAPH OF A GRAPH. Zahra Yarahmadi. Ali Reza Ashrafi. 1. Introduction
italian journal of pure and applied mathematics n. 33 2014 (101 106) 101 ON CHARACTERISTIC SUBGRAPH OF A GRAPH Zahra Yarahmadi Department of Mathematics, Faculty of Science Khorramabad Branch Islamic Azad
More informationGraph Algorithms. Vertex Coloring. Graph Algorithms
Graph Algorithms Vertex Coloring Graph Algorithms The Input Graph G = (V, E) a simple and undirected graph: V : a set of n vertices. E: a set of m edges. A A B C D E F C F D E B A 0 1 1 1 0 0 B 1 0 1 0
More informationMAD 3105 PRACTICE TEST 2 SOLUTIONS
MAD 3105 PRACTICE TEST 2 SOLUTIONS 1. Define a graph G with V (G) = {a, b, c, d, e}, E(G) = {r, s, t, u, v, w, x, y, z} and γ, the function defining the edges, is given by the table ɛ r s t u v w x y z
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 informationBegin at the beginning, the King said, gravely, and go on till you come to the end; then stop. Lewis Carroll, Alice in Wonderland
1 Graph Theory Begin at the beginning, the King said, gravely, and go on till you come to the end; then stop. Lewis Carroll, Alice in Wonderland The Pregolya River passes through a city once known as Königsberg.
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 informationGraph Algorithms. Edge Coloring. Graph Algorithms
Graph Algorithms Edge Coloring Graph Algorithms The Input Graph A simple and undirected graph G = (V,E) with n vertices in V, m edges in E, and maximum degree. Graph Algorithms 1 Matchings A matching,
More informationForests 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).
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 informationMath 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
More informationHomework 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
More informationCourse Notes for CS336: Graph Theory
Course Notes for CS336: Graph Theory Jayadev Misra The University of Texas at Austin 5/11/01 Contents 1 Introduction 1 1.1 Basics................................. 2 1.2 Elementary theorems.........................
More information