Unless otherwise stated throughout this article graph refers to a finite simple graph. This is followed by two chapters on planar graphs and colouring, with special reference to the fourcolour theorem. The basis of graph theory is in combinatorics, and the role of graphics is only in visualizing things. Graph theory is a branch of mathematics started by euler 45 as early as 1736. The notes form the base text for the course mat62756 graph theory. Acurveorsurface, thelocus ofapoint whosecoordinates arethevariables intheequation of the locus. The dots are called nodes or vertices and the lines are called edges. The length of the lines and position of the points do not matter. Algebraic graph theory has close links with group theory.
Jun 12, 2014 this video gives an overview of the mathematical definition of a graph. A gentle introduction to graph theory basecs medium. E wherev isasetofvertices andeisamultiset of unordered pairs of vertices. A data structure that consists of a set of nodes vertices and a set of edges that relate the nodes to each other the set of edges describes relationships among the vertices.
An undirected graph g v,e consists of a set v of elements called vertices, and a multiset e repetition of. Graphs can be infinite or finite, but by convention. There are various types of graphs, each with its own definition. Find materials for this course in the pages linked along the left. The concept of graphs in graph theory stands up on some basic terms such as point, line, vertex, edge, degree of vertices, properties of graphs, etc. Basic definitions definition a graph g is a pair v, e where v is a. E such that for all v2v, vappears as the endpoint of exactly one edge of f. An undirected graph is connected if for every pair of nodes u and v, there is a path between u and v. For example, if a graph represents a road network, the weights. Introduction these brief notes include major definitions and theorems of the graph theory lecture held by prof. These are the most basic graph theoretic definitions and a wonderful starting point to dive into articles about graph theory. Spectral graph theory is the branch of graph theory that uses spectra to analyze graphs. This will help to follow the discussion given in rest of the document as well as for easy reference to the nomenclature used afterward. A finite simple graph is an ordered pair, where is a finite set and each element of is a 2element subset of v.
Graph theory fundamentals a graph is a diagram of points and lines connected to the points. Introduction to graph coloring the authoritative reference on graph coloring is probably jensen and toft, 1995. A graph is an ordered pair g v, e comprising a set v of vertices or nodes and a collection of pairs of vertices from v called edges of the graph. This video gives an overview of the mathematical definition of a graph. Also, jgj jvgjdenotes the number of verticesandeg jegjdenotesthenumberofedges. The graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence class es. In recent years, graph theory has established itself as an important mathematical tool in a wide variety of subjects, ranging from operational research and chemistry to genetics and linguistics, and from electrical engineering and geography to sociology and architecture. A graph with maximal number of edges without a cycle. Introduction to graph theory applications math section. Note that the connected components of a forest are trees. It took a hundred years before the second important contribution of kirchhoff 9 had been made for the analysis of. A graph in this context is made up of vertices also called nodes or points which are connected by edges also called links or lines. In particular, it involves the ways in which sets of points, called vertices, can be connected by lines or arcs, called edges. A graph consists of some points and lines between them.
The two graphs shown below are isomorphic, despite their different looking drawings. My aim is to help students and faculty to download study materials at one place. It has at least one line joining a set of two vertices with no vertex connecting itself. A related class of graphs, the double split graphs, are used in the proof of the strong perfect graph theorem. Characterizations of connectedness and separability pdf.
A graph with no cycle in which adding any edge creates a cycle. A set of graphs isomorphic to each other is called an isomorphism class of graphs. Some examples, car navigation system efficient database build a bot to retrieve info off www representing computational models 4. Graphs in this context differ from the more familiar coordinate plots that portray mathematical relations and functions. Basic definitions definition a graph g is a pair v, e where v is a finite set and e is a set of 2element subsets of v. The mainpurpose of this chapter is to collect basic notions of the graph theory in one place and to be consistent in terminology.
The set v is called the vertex set of g and the set e is called the edge set of g. A regular graph on an odd number of vertices is class two proof. Most standard texts on graph theory such as diestel, 2000,lovasz, 1993,west, 1996 have chapters on graph coloring some nice problems are discussed in jensen and toft, 2001. They contain an introduction to basic concepts and results in graph theory, with a special emphasis put on the networktheoretic circuitcut dualism. Graphs used to model pair wise relations between objects generally a network can be represented by a graph many practical problems can be easily represented in terms of graph theory. Eigenvalues of graphs is an eigenvalue of a graph, is an eigenvalue of the adjacency matrix,ax xfor some vector x adjacency matrix is real, symmetric. A graph with n nodes and n1 edges that is connected. In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. In a graph, no two adjacent vertices, adjacent edges, or adjacent regions are colored with minimum number of colors. The subject of graph theory had its beginnings in recreational math problems see number game, but it has grown into a significant area of mathematical research, with applications in chemistry, operations research, social sciences, and computer science. An ordered pair of vertices is called a directed edge.
Introduction to graph theory terminology and basic concepts. Graph is a mathematical representation of a network and it describes the relationship between lines and points. Rob beezer u puget sound an introduction to algebraic graph theory paci c math oct 19 2009 10 36. A graph is a symbolic representation of a network and of its connectivity. The objects of the graph correspond to vertices and the relations between them correspond to edges. Free graph theory books download ebooks online textbooks.
Most standard texts on graph theory such as diestel, 2000,lovasz, 1993,west, 1996 have chapters on graph coloring. Basic graph definitions a data structure that consists of a set of nodes vertices and a set of edges that relate the nodes to each other the set of edges describes relationships among the vertices. Introduction to graph theory 5th edition by robin j. In an undirected graph, an edge is an unordered pair of vertices. In these algorithms, data structure issues have a large role, too see e. Graph coloring is nothing but a simple way of labelling graph components such as vertices, edges, and regions under some constraints.
It took a hundred years before the second important contribution of kirchhoff 9 had been made for the analysis of electrical networks. A split graph is a graph whose vertices can be partitioned into a clique and an independent set. Show that the following are equivalent definitions for a tree. A graph with a minimal number of edges which is connected. History of graph theory basic concepts of graph theory graph representations graph terminologies different type of graphs 3. There are several variations, for instance we may allow to be infinite. Graph theorydefinitions wikibooks, open books for an open. Mar 20, 2017 a gentle introduction to graph theory. Basic concepts in graph theory the notation pkv stands for the set of all kelement subsets of the set v. Definitions for the decision 1 module of ocrs alevel maths course, final examinations 2018. It can be shown that a graph is a tree iff it is connected and mn1.
Pdf basic definitions and concepts of graph theory. A graph is depicted diagrammatically as a set of dots depicting vertices connected by lines or curves depicting edges. We now have all the basic tools of graph theory and may now proceed to formalize these notions into some algebraic setting. Graph theorydefinitions wikibooks, open books for an. Ulman acknowledge that fundamentally, computer science is a science of abstraction. Pdf basic definitions and concepts of graph theory vitaly. Computer scientists must create abstractions of realworld problems that can.
Cmput 672 graph finite, no loops or multiple edges, undirecteddirected. Graph theory has a lot of areas of applications both in mathematics and in everyday life in general. A graph structure can be extended by assigning a weight to each edge of the graph. This paper is an expository piece on edgechromatic graph theory. A graph g is connected if for any two vertices v and w, there exists a path in g beginning at v and ending at w. The erudite reader in graph theory can skip reading this chapter. Graphtheoretic applications and models usually involve connections to the real. Graph theory history the origin of graph theory can be traced back to eulers work on the konigsberg bridges problem 1735, which led to the concept of an. For basic definitions and terminologies we refer to 1, 4. It gives some basic examples and some motivation about why to study graph theory. Some basic facts about linear programming problems.
A graph digraph x consists of two sets v and e, where v, is a nonempty set, called the vertex set and e is called the. Feb 20, 2014 graphs used to model pair wise relations between objects generally a network can be represented by a graph many practical problems can be easily represented in terms of graph theory 4. Graph theory is a branch of mathematics concerned about how networks can be encoded, and their properties measured. It implies an abstraction of reality so it can be simplified as a set of linked nodes. The first of these chapters 14 provides a basic foundation course, containing definitions and examples of graphs, connectedness, eulerian and hamiltonian. Prerequisite graph theory basics set 1 a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense related. Definition of graph a graph g v, e consists of a finite set denoted by v, or by vg if one wishes to make clear which graph is under consideration, and a collection e, or eg, of unordered pairs u, v of distinct elements from v. We write vg for the set of vertices and eg for the set of edges of a graph g. A graph is a diagram of points and lines connected to the points. Mathematics graph theory basics set 2 geeksforgeeks. If the vertices of a graph can be divided into two sets a, b such that each edge connects a vertex from a and a vertex from b, the graph is called bipartite. You can read about these examples right here on the math section. Graphs with weights or weighted graphs are used to represent structures in which pair wise connections have some numerical values.
Terminology and representations of graphs techie delight. E where v or vg is a set of vertices eor eg is a set of edges each of which is a set of two vertices undirected, or an ordered pair of vertices directed two vertices that are contained in an edge are adjacent. This post discuss the basic definitions in terminologies associated with graphs and covers adjacency list and adjacency matrix representations of the graph data structure. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where. Here, in this chapter, we will cover these fundamentals of graph theory. Graph theory, branch of mathematics concerned with networks of points connected by lines. The opening chapters provide a basic foundation course, containing definitions and examples, connectedness, eulerian and hamiltonian paths and cycles, and trees, with a range of applications.
630 45 1258 693 70 816 835 876 713 1548 1445 1202 447 1038 1212 225 264 110 1557 1152 1119 300 927 1296 1496 1067 814 1004 1140 499 811 1396