GraphTheory&Probability/ GraphTheory DMTH501/DMTH601 GRAPH THEORY AND PROBABILITY/ / GRAPH THEORY Copyright © 2011 C Anuranjan Misra All rights reserved Produced & Printed by EXCEL BOOKS PRIVATE LIMITED A-45, Naraina, Phase-I, New Delhi-110028 for Lovely Professional University Phagwara SYLLABUS DMTH501 Graph Theory and Probability Objectives: To learn the fundamental concept in graph theory and probabilities, with a sense of some of its modern application. Also to learn, understand and create mathematical proof, including an appreciation of why this is important. After the completion of this course prospective students will be able to tackle mathematical issues to applicable in data structure related course. Sr. No. Description 1 Graph Theory: Graph and sub graphs, Isomorphic, homomorphism graphs, Paths, Hamiltonian Circuits, Eulerian Graph, Connectivity 2 The Bridges of konigsberg,Transversal, Multi graphs, Labelled graph, Complete, regular and bipartite graphs, planar graphs 3 Graph colourings, Chromatic number, Connectivity, Directed graphs, basic definitions, Tree graphs,Binary trees, rooted trees 4 minimal spanning tree, Prim’s algorithm, shortest path. 5 Propositions and compound propositions,Basic logical operations, Truth tables, Tautologies and Contradictions,Logical equivalence 6 Logic gates, Logic circuits, and switching function, Partially ordered set 7 Lattice, Boolean algebra,Lattice as Boolean algebra, Application of Boolean algebra to on –off switching theory. 8 Sample space, events, and probability functions, Examples using counting methods, sampling with or without replacement, Algebra of events 9 Conditional probability, partitions of sample space. Theorem of total probability, Baye’s theorem, independence, Random variables 10 Probability mass functions,Discrete distributions: - Binomial, Poisson, geometric. Expectation: - mean and variance DMTH601 Graph Theory 1 Graph Theory: Graph and sub graphs, Isomorphic, homomorphism graphs, 2 Paths, Hamiltonian circuits, Eulerian graph, connectivity 3 The bridges of Konigsberg, Transversal, Multi graphs, Labeled graph 4 Complete, regular and bipartite graphs, planar graphs 5 Graph colorings, Chromatic number, Connectivity, Directed graphs 6 Basic definitions , Tree graphs , Binary trees, rooted trees 7 Minimal spanning tree, Prim’s algorithm, shortest path 8 Propositions and compound propositions, 9 Basic logical operations, Truth tables, Tautologies and Contradictions, Logical equivalence 10 Logic gates, Logic circuits, and switching function, Partially ordered set CONTENTS Unit 1: Introduction to Graph Theory 1 Unit 2: Types of Graphs 15 Unit 3: Eulerian and Hamiltonian Graphs 29 Unit 4: Graphs Colouring 39 Unit 5: Tree Graphs 52 Unit 6: Algorithm 70 Unit 7: Boolean Algebra 86 Unit 8: Mathematical Logic 115 Unit 9: Hasse Diagrams and Posets 138 Unit 10: Supremum and Infimum 152 Unit 11: Probability Theory 176 Unit 12: Conditional Probability 191 Unit 13: Random Variables 210 Unit 14: Probability Distributions 250 Corporate and Business Law 6 LOVELY PROFESSIONAL UNIVERSITY Unit 1: Introduction to Graph Theory Unit 1: Introduction to Graph Theory Notes CONTENTS Objectives Introduction 1.1 Related Terms 1.2 Multigraph 1.3 Subgraph 1.4 Homomorphism Graphs 1.5 Path 1.6 Connectivity 1.6.1 Definitions of Connectivity 1.6.2 Menger’s Theorem 1.6.3 Computational Aspects 1.6.4 Bounds on Connectivity 1.6.5 Other Properties 1.7 Summary 1.8 Keywords 1.9 Self Assessment 1.10 Review Questions 1.11 Further Readings Objectives After studying this unit, you will be able to: Define graph Evaluate the terms of graphs Describe the different types of path Explain homomorphism graphs Describe the paths and connectivity LOVELY PROFESSIONAL UNIVERSITY 1 Graph Theory and Probability Notes Introduction The word graph refers to a specific mathematical structure usually represented as a diagram consisting of points joined by lines. In applications, the points may, for instance, correspond to chemical atoms, towns, electrical terminals or anything that can be connected in pairs. The lines may be chemical bonds, roads, wires or other connections. Applications of graph theory are found in communications, structures and mechanisms, electrical networks, transport systems, social networks and computer science. 1.1 Related Terms A graph is a mathematical structure comprising a set of vertices, V, and a set of edges, E, which connect the vertices. It is usually represented as a diagram consisting of points, representing the vertices (or nodes), joined by lines, representing the edges (Figure 1.1). It is also formally denoted by G (V, E). Figure 1.1 Vertices are incident with the edges which joins them and an edge is incident with the vertices it joins. The degree of a vertex v is the number of edges incident with v, Loops count as 2. The degree sequence of a graph G is the sequence obtained by listing, in ascending order with repeats, the degrees of the vertices of G (e.g. in Figure 1.2 the degree sequence of G is (1, 2, 2, 3, 4). The Handshaking Lemma states that the sum of the degrees of the vertices of a graph is equal to that twice the no. of edge this follow reality from the fact that each edge join two vertices necessarily distinct) and so contributes 1 to the degree of each of those vertices. A walk of length k in a graph is a succession of k edges joining two vertices. NB Edges can occur more than once in a walk. A trail is walk in which all the edges (but not necessarily all the vertices) are distinct. A path is a walk in which all the edges and all the vertices are distinct. So, in Figure 1.2, abdcbde is a walk of length 6 between a and e. It is not a trail (because edge bd is traversed twice). The walk adcbde is a trail length 5 between a and e. It is not a path (because vertex d is visited twice). The walk abcde is a path of length 4 between a and e. Figure 1.2 2 LOVELY PROFESSIONAL UNIVERSITY Unit 1: Introduction to Graph Theory Notes Did u know? What is closed walk? A closed walk or trail is a walk or trail starting and ending at the same vertex. Digraphs Digraphs are similar to graphs except that the edges have a direction. To distinguish them from undirected graphs the edges of a digraph are called arcs. Much of what follows is exactly the same as for undirected graphs with ‘arc’ substituted for ‘edge’. We will attempt to highlight the differences. A digraph consists of a set of elements, V, called vertices, and a set of elements, A, called arcs. Each arc joins two vertices in a specified direction. Two or more arcs joining the same pair of vertices in the same direction are called multiple arcs (see Figure 1.3). NB two arcs joining the same vertices in opposite directions are not multiple arcs. An arc joining a vertex to itself is a loop (see Figure 1.3). Figure 1.3 Figure 1.4 Figure 1.5 A digraph with no multiple arcs and no loops is a simple digraph (e.g. Figure 1.4 and Figure 1.5). Two vertices joined by an arc in either direction are adjacent. Vertices are incident to and form the arc which joins them and an arc is incident to and from the vertices its joins. Two digraphs G and H are isomorphic if H can be obtained by relabelling the vertices of G, i.e. there is a one-one correspondence between the vertices of G and those of H such that the number and direction of the arcs joining each pair of vertices in G is equal to the number and direction of the arc joining the corresponding pair of vertices in H. A subdigraph of G is a digraph all of whose vertices and arcs are vertices and arcs of G. The underlying graph of a digraph is the graph obtained by replacing of all the arcs of the digraph by undirected edges. The out-degree of a vertex v is the number of arcs incident from v and the in-degree of a vertex v is the number of arcs incident to v. Loops count as one of each. The out-degree sequence and in-degree sequence of a digraph G are the sequences obtained by listing, in ascending order with repeats, the out-degrees and in-degrees of the of the vertices of G. The Handshaking Lemma states that the sum of the out-degrees and of the in-degrees of the vertices of a graph are both equal to the number of arcs. This is pretty obvious since every arc contributes one to the out-degree of the vertex from which it is incident and one to the in-degree of the vertex to which it is incident. A walk of length k in a digraph is a succession of k arcs joining two vertices. LOVELY PROFESSIONAL UNIVERSITY 3 Graph Theory and Probability Notes A trail is a walk in which all the arcs (but not necessarily all the vertices) are distinct. A path is a walk in which all the arcs and all the vertices are distinct. A connected digraph is one whose underlying graph is a connected graph. A disconnected digraph is a digraph which is not connected. A digraph is strongly connected if there is a path between every pair of vertices. Notice here we have a difference between graphs and digraphs. The underlying graph can be connected (a path of edges exists between every pair of vertices) whilst the digraph is not because of the directions of the arcs (see Figure 1.5 for a graph which is connected but not strongly connected). A closed walk/trail is a walk/tail starting and ending at the same vertex. A cycle is a closed path, i.e. a path starting and ending at the some vertex.