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© 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?