Graphs

10 Key excerpts on "Graphs"

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  Discrete Mathematics for Computing

  • Peter Grossman(Author)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  Introduction to graph theory 10.1 What is a graph? The objects that we study in the branch of mathematics known as graph theory are not Graphs drawn with x and y axes. In this chapter, the word ‘graph’ refers to a structure consisting of points (called ‘vertices’), some of which may be joined to other vertices by lines (called ‘edges’) to form a network. Structures of this type abound in computing. The computers on a site may be connected into a local area network, which in turn may be linked to wider networks, including the Internet. The circuitry inside a computer (which we represented schematically by digital circuit diagrams in Chapter 8) provides another example of a graph or network structure. At a more abstract level, we saw in Chapter 5 how a relation on a set can be depicted using a diagram that takes the form of a graph. There is a particular type of graph called a tree , which we will study in Chapter 11. Trees are used in computing to represent the structure of expressions; we saw an example of a logical expression tree in Chapter 4. Trees can also be used to represent decision processes, in which we are faced with two or more choices at each step of a procedure; the tree diagrams in Chapter 9 were of this type. In this chapter, we introduce Graphs, study their basic properties, and investigate some practical problems in which they can be applied. In Chapter 11, we will study trees and investigate some of their applications to computing. 10.2 Basic concepts in graph theory Definition A graph consists of a non-empty set of points, called vertices (singular: vertex ), and a set of lines or curves, called edges , such that every edge is attached at each end to a vertex. 180 CHAPTER 10 An example of a graph is shown in Figure 10.1, with the edges and vertices labelled. An edge is said to be incident to the vertices to which it is attached. For example, the edge e 1 is incident to the vertices v 1 and v 2 in the graph in Figure 10.1.

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  Fundamentals of Graph Theory

  • Allan Bickle(Author)
  • 2020(Publication Date)
  • American Mathematical Society

    (Publisher)

  Definition 1.1. A graph G is a mathematical object consisting of a finite non-empty set of objects called vertices V ( G ) (the vertex set ), and a set of edges E ( G ) (the edge set ). An edge is two-element subset of the vertex set. We commonly use G and H for Graphs; u , v , w, . . . , for vertices; and e and f for edges. An edge e = { u, v } will typically be written uv or vu , dropping the inconvenient braces. The name “graph” should not be confused with the graph of a function, an unrelated mathematical concept. The term “network”, which is common in com-puter science and technology, would probably be a more intuitive name. However, “graph” is now standard in mathematics. The terms “vertex” and “edge” come from geometry, as they can be used to represent the geometric objects with the same names. Several variations on the concept of a graph are possible, allowing multiple edges or directed edges. If we allow multiple edges between vertices, the edges must be in a multiset , which allows multiple copies of the same object. For directed edges, we replace unordered pairs with ordered pairs of vertices. Definition 1.2. A multigraph G is a mathematical object consisting of a finite nonempty set of objects called vertices V ( G ) and a multiset E ( G ) of pairs of vertices. A directed graph (digraph) D is a mathematical object consisting of a finite nonempty set of objects called vertices V ( D ) and a set E ( D ) of ordered pairs of distinct vertices called directed edges . A directed multigraph replaces the set E ( D ) with a multiset. Examples of a multigraph and a digraph are shown above. Every graph is also a multigraph. A multigraph is allowed, but not required, to have multiple edges between pairs of vertices. 1.1. Graphs as Models 3 Each of these mathematical objects can be used to model many real-world situations. Which one is chosen depends on whether multiple or directed edges make sense in the context of the problem.

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  Computational Discrete Mathematics

  Combinatorics and Graph Theory

  • (Author)
  • 2014(Publication Date)
  • Orange Apple

    (Publisher)

  This latter type of graph is called a directed graph and the edges are called directed edges or arcs ; in contrast, a graph where the edges are not directed is called undirected . Vertices are also called nodes or points , and edges are also called lines . Graphs are the basic subject studied by graph theory. The word graph was first used in this sense by James Joseph Sylvester in 1878. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining Graphs and related mathematical structures. Graph A general example of a graph (actually, a pseudograph) with three vertices and six edges In the most common sense of the term, a graph is an ordered pair G = ( V , E ) comprising a set V of vertices or nodes together with a set E of edges or lines , which are 2-element subsets of V (i.e., an edge is related with two vertices, and the relation is represented as unordered pair of the vertices with respect to the particular edge). To avoid ambiguity, this type of graph may be described precisely as undirected and simple . Other senses of graph stem from different conceptions of the edge set. In one more generalized notion, E is a set together with a relation of incidence that associates with each edge two vertices. In another generalized notion, E is a multiset of unordered pairs of (not necessarily distinct) vertices. Many authors call this type of object a multigraph or pseudograph. All of these variants and others are described more fully below. The vertices belonging to an edge are called the ends , endpoints , or end vertices of the edge. A vertex may exist in a graph and not belong to an edge. ________________________ WORLD TECHNOLOGIES ________________________ V and E are usually taken to be finite, and many of the well-known results are not true (or are rather different) for infinite Graphs because many of the arguments fail in the infinite case. The order of a graph is | V | (the number of vertices).

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  Complex Networks

  Principles, Methods and Applications

  • Vito Latora, Vincenzo Nicosia, Giovanni Russo(Authors)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 Graphs and Graph Theory Graphs are the mathematical objects used to represent networks, and graph theory is the branch of mathematics that deals with the study of Graphs. Graph theory has a long his- tory. The notion of the graph was introduced for the first time in 1763 by Euler, to settle a famous unsolved problem of his time: the so-called Königsberg bridge problem. It is no coincidence that the first paper on graph theory arose from the need to solve a problem from the real world. Also subsequent work in graph theory by Kirchhoff and Cayley had its root in the physical world. For instance, Kirchhoff’s investigations into electric circuits led to his development of a set of basic concepts and theorems concerning trees in Graphs. Nowa- days, graph theory is a well-established discipline which is commonly used in areas as diverse as computer science, sociology and biology. To give some examples, graph theory helps us to schedule airplane routing and has solved problems such as finding the max- imum flow per unit time from a source to a sink in a network of pipes, or colouring the regions of a map using the minimum number of different colours so that no neighbouring regions are coloured the same way. In this chapter we introduce the basic definitions, set- ting up the language we will need in the rest of the book. We also present the first data set of a real network in this book, namely Elisa’s kindergarten network. The two final sections are devoted to, respectively, the proof of the Euler theorem and the description of a graph as an array of numbers. 1.1 What Is a Graph? The natural framework for the exact mathematical treatment of a complex network is a branch of discrete mathematics known as graph theory [48, 47, 313, 150, 272, 144]. Dis- crete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete, i.e. made up of distinct parts, not supporting or requiring the notion of continuity.

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  The Mathematics of Finite Networks

  An Introduction to Operator Graph Theory

  • Michael Rudolph(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  On its most basic conceptual level, a network, or graph in the mathematical literature, is comprised of a collection of discrete objects, interchangeably called vertices or nodes, and a collection of edges, also called links or bonds, which describe relations between these objects. Remaining with the example of K¨ onigsberg as the iconic city whose now historical mystery 13 14 Classical Graph Theory inspired the mathematical field of network studies, or graph theory, we find that each of the city’s four river banks and islands constitutes a node, while the seven bridges connecting these areas establish relations between these nodes and, thus, can be viewed as comprising a set of edges. The collection of areas and bridges then forms the graph of K¨ onigsberg’s bridges, as illustrated in Fig. 1.1. This brilliant insight of Leonard Euler, by abstracting from colourful yet less relevant details, not only allowed for a new way to look at and eventu- ally solve the historical problem associated with K¨ onigsberg’s seven bridges itself, but indeed established a new powerful framework which matured since Euler’s time into an indispensable tool for studying the governing structural and functional characteristics of a sheer unlimited range of real-world phenom- ena and physical systems. In order to fully realise and harness the power of this framework beyond a mere descriptive vessel, however, we certainly require a mathematically more rigorous representation of a network and its conceptual makeup. To that end, we formulate Definition 2.1 (finite graph) A finite network or graph G is an ordered tuple G = (N , E) of a set N , ∅ of nodes i with cardinality |N| ∈ N : |N| < ∞, and a set E of edges with cardinality |E| ∈ N 0 comprised of ordered pairs (i, j) ∈ N × N , each denoting an adjacency relation, or link, between a source node i and a target node j.

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  Software Testing

  • Yogesh Singh(Author)
  • 2011(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Essentials of Graph Theory Graph theory has been used extensively in computer science, electrical engineering, communication systems, operational research, economics, physics and many other areas. Any physical situation involving discrete objects may be represented by a graph along with their relationships amongst them. In practice, there are numerous applications of Graphs in modern science and technology. Graph theory has recently been used for representing the connectivity of the World Wide Web. Global internet connectivity issues are studied using Graphs like the number of links required to move from one web page to another and the links which are used to establish this connectivity. It has also provided many ways to test a program. Some testing techniques are available which are based on the concepts of graph theory. 3.1 WHAT IS A GRAPH? A graph has a set of nodes and a set of edges that connect these nodes. A graph G = (V, E) consists of a non-empty finite set V of nodes and a set E of edges containing ordered or unordered pairs of nodes. V = (n 1 , n 2 , n 3 ……..n m ) and E = (e 1 , e 2 , e 3 ………e k ) If an edge e i E is associated with an ordered pair or an unordered pair (n i , n j ), where n i , n j V, then the e i is said to connect the nodes n i and n j . The edge e i that connects the node n i and n j is called incident on each of the nodes. The pair of nodes that are connected by an edge are called adjacent nodes. A node, which is not connected to any other node, is called an isolated node. A graph with only isolated nodes is known as null graph. If in graph G = (V, E), each edge e i E is associated with an ordered pair of nodes, then graph G is called a directed graph or digraph. If each edge is associated with an unordered pair of nodes, then a graph G is called an undirected graph.

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  Discrete Mathematics

  Proofs, Structures and Applications, Third Edition

  • Rowan Garnier, John Taylor(Authors)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  Like many of the very great mathematicians of his era, Euler contributed to almost every branch of pure and applied mathematics. He is also responsible, more than any other person, for much of the mathematical notation in use today. 548 Definitions and Examples 549 two quite different meanings, although it is usually clear from the context which meaning is intended. What, then, is a ‘graph’? Intuitively, a graph is simply a collection of points, called ‘vertices’, and a collection of lines, called ‘edges’, each of which joins either a pair of points or a single point to itself. A familiar example, which serves as a useful analogy, is a road map which shows towns as vertices and the roads joining them as edges. For mathematical purposes we require a more precise definition. In order to define a graph, we first need to specify the set of its vertices and the set of its edges. Then we need to say, in precise mathematical terms, which edges join which vertices. An edge is defined as having a vertex at each end, so we need to associate with every edge of the graph its endpoint vertices. The endpoints of an edge are either a pair of vertices (if the edge joins two different vertices) or a single vertex (if the edge joins a vertex to itself). Thus for every edge e of a graph we define a set { v 1 , v 2 } of vertices which specifies that e joins vertices v 1 and v 2 , where of course we need to allow the possibility that v 1 = v 2 . Now this set { v 1 , v 2 } , which we denote by δ ( e ) , is a subset of the set of vertices. Therefore δ ( e ) is an element of the power set of the vertex set. This leads us to the following formal definition. Its rather technical nature should not be allowed to obscure the essentially simple concept that is being described.

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  Algebraic Graph Theory

  Morphisms, Monoids and Matrices

  • Ulrich Knauer, Kolja Knauer(Authors)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  1 Directed and undirected Graphs In this chapter, we collect some important basic concepts. These concepts are essential for all mathematical modeling based on Graphs. The language and visual representa-tions of Graphs are such powerful tools that graph models can be encountered almost everywhere in mathematics and informatics, as well as in many other fields. The most obvious phenomena that can be modeled by Graphs are binary relations. Moreover, Graphs and relations between objects in a formal sense can be considered the same. The concepts of graph theory also play a key role in the language of category theory, where we consider objects and morphisms. It is not necessary to read this chapter first. Anybody who is familiar with the basic notions may just refer back to this chapter as needed for a review of notation and concepts. 1.1 Formal description of Graphs We shall use the word “graph” to refer to both directed and undirected Graphs. Only when discussing concepts or results that are specific to one of the two types of graph we will use the corresponding adjective explicitly. Definition 1.1.1. A directed graph or digraph or also oriented graph is a triple G = ( V , E , p ) where V and E are sets and p : E → V 2 is a mapping. We call V the set of vertices or points and E the set of edges or arcs of the graph. Sometimes we will write these sets as V ( G ) and E ( G ) . The mapping p is called the incidence mapping . The mapping p defines two more mappings o , t : E → V by ( o ( e ), t ( e )) := p ( e ) ; these are also called incidence mappings. We call o ( e ) ∈ V the origin or source and t ( e ) ∈ V the tail or end of e ∈ E . As p defines the mappings o and t , these in turn define p by p ( e ) := ( o ( e ), t ( e )) . We will mostly be using the first of the two alternatives G = ( V , E , p ) or G = ( V , E , o , t ). We say that the vertex x and the edge e are incident if x is the source or the tail of e .

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  Computational Topology

  An Introduction

  • Herbert Edelsbrunner, John L. Harer(Authors)
  • 2009(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter I Graphs In topology we think of a graph as a 1-dimensional geometric object, vertices being points and edges being curves connecting these points in pairs. This view is dif-ferent from but compatible with the interpretation of a graph common in discrete mathematics where the vertices are abstract elements and the edges are pairs of these elements. In more than one way, this book lives in the tension between the discrete and the continuous, and Graphs are just one example of this phenomenon. We begin with the discussion of an intrinsic property, namely whether a graph is connected or not. Indeed, this does not depend on where we draw the graph, on paper or in the air. Following are extrinsic considerations about curves and Graphs in the plane and in 3-dimensional space. While the extrinsic questions are natural to most people, the mathematician usually favors the intrinsic point of view since it tends to lead to more fundamental insights of more general validity. I.1 Connected Components A theme that goes through this entire book is the exchange between discrete and continuous models of reality. In this first section, we compare the notion of con-nectedness in discrete Graphs and continuous spaces. Simple Graphs. An abstract graph is a pair G = ( V, E ) consisting of a set of vertices , V , and a set of edges , E , each a pair of vertices. We draw the vertices as points or little circles and the edges as line segments or curves connecting the points. The graph is simple if the edge set is a subset of the set of unordered pairs, E ⊆ ( V 2 ) , which means that no two edges connect the same two vertices and no edge joins a vertex to itself. For n = card V vertices, the number of edges is m = card E ≤ ( n 2 ) . Every simple graph with n vertices is a subgraph of the complete graph , K n , that contains an edge for every pair of vertices; see Figure I.1. 3 4 I Graphs Figure I.1: The complete graph with five vertices, K 5 .

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  Discrete Structures and Their Interactions

  • Jason I. Brown(Author)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Chapter 3 Graphs and Directed Graphs We begin by discussing some theory of Graphs and diGraphs. Before we do, we shall remind ourselves of some standard notation. The complete graph K n on V is any simple graph of order n with all ( n 2 ) edges. The empty graph E n is any graph of order n with no edges. The path P n of order n has vertex set [ n ] with edges { i,i +1 } for i = 0 ,...,n − 2. The cycle C n of order n has vertex set [ n ] with edges { i,i +1 } for i = 1 ,...,n , with addition modulo n (a cycle of order 1 is a loop, while a cycle of order 2 consists of two parallel edges). The complete k –partite graph K l 1 ,l 2 ,...,l k has vertex set V = V 1 ∪···∪ V k , where the V i ’s are disjoint and | V i | = l i , and edge set { uv : u ∈ V i , v ∈ V j , i negationslash = j } . In the case k = 2, we say bipartite rather than 2–partite. The stars are the Graphs K 1 ,l . A forest is a graph that contains no cycles, and a tree is a connected forest. There are no more important representations of Graphs and diGraphs than the standard geometric ones. A geometric representation of a directed graph D = ( V,A ) is a collection of continuous functions ρ : V → R 2 and φ a : [0 , 1] → R 2 , for all a ∈ A , such that ρ is 1–1, and for each a = ( u,v ) ∈ A , φ a is a Jordan arc or curve (i.e. φ a restricted to [0 , 1) is 1–1), φ a (0) = ρ ( u ), φ a (0) = ρ ( v ) FIGURE 3.1: A geometric representation of a digraph 23 24 Discrete Structures and Their Interactions and φ a ((0 , 1)) ∩{ ρ ( w ) : w ∈ V } = ∅ . If moreover φ a ((0 , 1)) ∩ φ b ((0 , 1)) = ∅ for distinct arcs a and b , then the representation is proper ). The image of the maps is what is usually drawn “as” the directed graph, with arrows on each arc indicating the direction (i.e. if a = ( u,v ), then the arrow on φ a ([0 , 1]) points towards φ a ( v )). Example: Consider V = { 1 , 2 , 3 , 4 } and A = { (1 , 1) , (1 , 2) , (2 , 4) , (4 , 2) , (3 , 4) } .

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