© Nuffield Foundation 2012

Nuffield Free-Standing Mathematics Activity

Networks

© Rudolf Stricker

NetworksThis diagram shows a network.

Problems involving the study of networks include optimisation tasks such as finding the shortest route and minimum cost.

This activity introduces some of the terms and methods used.

vertex (node)

edge

Vertices can be even or odd

or undirected (two-way)

Edges can be directed (one-way)

This is an example of a graph.

This graph is connected

Think about …Why must the sum of the degrees of the vertices in any graph always be even? What can you say about the number of odd vertices in a graph?

The degree of a vertex is the number of edges that meet there

49 mins

39 mins

28 mins

22 mins

24 mins

29 mins

34 mins

15 miles

17 miles

15 miles

12 miles

17 miles

21 miles

20 miles

Networks

Colchester

Stowmarket

IpswichSudbury

Bury St Edmunds

Harwich

A network is a weighted graph.

The weights could be distances, times or costs.

Paths

Paths are routes that do not visit any vertex more than onceand do not go along any edge more than once.

A cycle forms a loop by returning to its starting point.

Colchester

Stowmarket

IpswichSudbury

Bury St Edmunds

Harwich

Adjacency matrix

010001100110000100011010010101100010

B Su C H I St BSuC HISt

Colchester

Stowmarket

IpswichSudbury

Bury St Edmunds

Harwich

Distance matrixB Su C H I St

BSuC HISt 15 miles

17 miles

15 miles

12 miles

17 miles

21 miles

20 miles

01200015120017200000210001721015002001501715000170

Think about…What do you notice about the pattern in these matrices?

Reflect on your work• Explain what is meant by the following terms:

network, graph, edge, vertex, node, degree, directed, undirected, weighted, path, cycle, connected, adjacency matrix, distance matrix .

© Nuffield Foundation 2012

• If a road is one way, then when planning a driving route it may be possible to get from A to B, but not from B to A.What difference would this make to the adjacency matrix?