Networks & Matrices (Edexcel IAL Maths)

Introduction to Networks & Matrices

Spec point — 2.1

What is a network?

• A network is a weighted graph

  • A network is used to model real-world situations
  • The weight of an edge usually represents a measure such as distance or time

What is a matrix?

• A matrix can be used to represent a graph or network in the form of a table

  • The elements within the matrix give information about the connections between the different vertices

• A matrix is typically set up with the following labelling system

  • Rows are labelled with the 'from' vertices
  • Columns are labelled with the 'to' vertices

Networks & Matrices

Spec point — 2.1

What is an adjacency matrix?

• An adjacency matrix is a square matrix

  • All of the vertices in the graph are listed as the headings for both the rows and columns
  • The leading diagonal is the the line going from the top left cell to the bottom right cell

• An adjacency matrix can be used to show the number of direct connections between two vertices

• An entry of 1 in the matrix means that there is a direct connection (i.e. an edge) between that pair of vertices

• An entry of 0 in the matrix means that there is no direct connection between that pair of vertices

• In a simple graph the only entries are 0 or 1

• A loop is indicated in an adjacency matrix with a value in the leading diagonal

  • In an undirected matrix the value in the leading diagonal will be 2

    • You can use the loop to travel out of and into the vertex in two different directions

  • In a directed matrix, if the loop has direction, the value in the leading diagonal will be 1

    • You can only travel along the loop out of and back into the vertex in one direction
    • An undirected loop in a directed matrix will still have a value of 2

  • For a graph with no loops every entry in the leading diagonal will be 0

• An undirected graph is symmetrical about the leading diagonal

  • A directed graph is not symmetrical about the leading diagonal

What is a distance matrix?

• A distance matrix is different to an adjacency matrix

  • The value in each cell is the weight of the edge connecting that pair of vertices
  • Weight could be cost, distance, time etc.

• An empty cell is used to indicate that there is no connection (i.e. no edge) between a pair of vertices

• An undirected network is symmetrical about the leading diagonal

  • A directed network is not symmetrical about the leading diagonal

• When drawing a network from its distance matrix be careful when labelling the edges

  • For an undirected graph the two cells between a pair of vertices will be the same

    • Connect the vertices with one edge labelled with the relevant weight

  • For a directed graph where two cells between a pair of vertices have different values

    • Draw two edges between the vertices
    • Label each with the correct weight and direction

• A distance matrix can be used to work out the weight of different walks in the network

Worked Example a) Write down the adjacency matrix for the graph shown below.

Answer:

$\begin{gathered} \begin{gathered} A B C D E \end{gathered} \\ \begin{gathered} A \\ B \\ C \\ D \\ E \end{gathered} & ( \begin{gathered} 0 1 1 0 1 \\ 1 0 0 0 0 \\ 1 0 1 1 1 \\ 0 0 0 0 1 \\ 2 1 1 1 2 \end{gathered} ) \end{gathered}$

b) The table below shows the time taken in minutes to travel by car between 4 different towns.

	A	B	C	D
A		16	35
B	16		20	18
C	35	20		34
D		23	34

Draw the network described by this distance matrix.

Answer: