Nuffield Free-Standing
Mathematics Activity
Networks
© Rudolf Stricker
© Nuffield Foundation 2012
Networks
This 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.
This is an example
of a graph.
edge
Edges can be directed
(one-way)
or undirected (two-way)
vertex (node)
The degree of a vertex
is the number of edges
that meet there
Vertices can be
even or odd
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?
Networks
A network is a weighted graph.
The weights could be distances, times or costs.
Bury St
Edmunds
15 miles
22 mins
17 miles
28 mins
Sudbury
Stowmarket
20 miles 49 mins
15 miles
39 mins
Colchester
12 miles
24 mins
Ipswich
17 miles
29 mins
21 miles
34 mins
Harwich
Paths
Paths are routes that do not visit any vertex more than once
and do not go along any edge more than once.
A cycle forms a loop
by returning to its
starting point.
Adjacency matrix
B
Su
C
H
I
St
Bury St
Edmunds
Stowmarket
Ipswich
Sudbury
Colchester
Harwich
B
0

1
0

0

0
1

Su
1
0
1
0
1
0
C
0
1
0
1
1
0
H
0
0
1
0
0
0
I
0
1
1
0
0
1
St
1

0
0

0

1
0 
Distance matrix
Bury St Edmunds
B
Stowmarket
Su
17 miles
12 miles
C
H
20 miles
Ipswich
Sudbury
I
St
17 miles
15 miles
15 miles
B
0

17
0

0

0
15

Su C H
17 0 0
0 15 0
15 0 21
0 21 0
20 17 0
0
0
0
Colchester
21 miles
Harwich
Think about…
What do you notice about the pattern in these matrices?
I St
0 15 

20 0 
17 0 

0 0

0 12 
12 0 
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 .
• 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?
© Nuffield Foundation 2012