Graph Theory (Edexcel IAL Maths)

Introduction to Graph Theory

What is graph theory?

• Graph theory is the study of graphs, which are mathematical structures used to represent objects and the connections between them

  • Graphs can be used in modelling many real-life applications, e.g. electrical circuits, flight paths, maps etc.

Edges (Arcs) & Vertices (Nodes)

Spec point — 2.1

What are the different parts of a graph?

• A graph is made up of a number of points (vertices or nodes) that are connected by lines (edges or arcs)

• A vertex (node) can represent an object, a place or a person

  • Adjacent vertices are connected by an edge

  • Vertices are usually labelled with a letter, e.g. A, B, C...

    • The list of vertices in a graph is sometimes called the vertex set

• An edge (arc) forms a connection between two vertices

  • Edges are described by the nodes they connect, e.g. AB, AC, BE...

    • The list of edges in a graph is sometimes called the edge set

  • Adjacent edges share a common vertex

  • There may be multiple edges connecting two vertices

  • An edge that starts and ends at the same vertex is called a loop

• Typically a graph will be drawn such that edges do not overlap

  • Note that if a graph is drawn with overlapping edges, the edges are not connected at points where they overlap
  • Edges are only connected at vertices

Properties of Graphs

Spec point — 2.1

What are the properties of graphs?

• The edges of a graph may be assigned a numerical value

  • This value is called the weight (of an edge)
  • Weight is often a measure such as distance, time or money

• A walk is a finite series of edges in a graph that starts at one vertex and moves from vertex to vertex

  • The (total) weight of a walk is the sum of the weights of the edges that it consists of

• A path is a walk where no vertex is visited more than once

• A trail is a walk where no edge is visited more than once

  • Every path is also a trail but not every trail is a path

• A cycle (or circuit) is a path that starts and finishes at the same vertex

  • It is also known as a closed path

• A tour is a walk that visits every vertex and returns to its start vertex

Types of Graph

Spec point — 2.1

What are the types of graph?

• A connected graph is a graph in which all of its vertices are connected to each other

  • Two vertices are connected if there is a path between them

    • There does not need to be an edge connecting the pair of vertices

• A complete graph is a graph in which each vertex is connected by an edge to each of the other vertices

  • A complete graph with n vertices is labelled K_n

• If the edges of a graph have a weight, the graph is known as a weighted graph (or network)

  • Networks are not usually drawn to scale

• If the edges of a graph are assigned a direction, they are known as directed edges and the graph is known as a digraph

  • Directed edges of a graph can only be traversed in the direction indicated

• A simple graph is undirected and unweighted and contains no loops or vertices connected by multiple edges
• Given a graph G, a subgraph is a graph which contains only edges and vertices that appear in G
• A tree is a connected graph that does not contain any cycles
• A spanning tree is a subgraph (of a graph G) which is also a tree, and which contains all the vertices from G

• Isomorphic graphs are graphs which show the same information but which are drawn in different ways

  • The number of vertices and the number of edges connected to each vertex are the same

Exam Hint

• There are a lot of specific terms involved in graph theory

  • You are often asked to describe these terms in an exam, so make sure you learn the definitions
  • Note that in many cases there are two terms that describe the same thing!

• Make sure that any graphs you draw are big and clear so they are easy for the examiner to read

Worked Example The graph G shown below is a connected, unweighted, directed graph with 5 vertices.

a) Write down a cycle, starting at vertex A and visiting exactly 2 other vertices.

Answer:

A cycle starts and ends at the same vertex, with no other vertex visited more than once Two other vertices (from the 4 possibilities) need to be included The directed edges mean, in this case, there is only one such cycle

Final answer: ADCA

b) Write down a path, consisting of more than one edge, that starts at vertex A and ends at vertex B.

Answer:

A path is a walk that does not repeat any nodes

*You must only traverse edges in the directions indicated

Final answer: ADCEB

A path does not have to visit every node (so ACEB or ABEB, for example, are also acceptable)