Mathematics Standard 1 Year 12
Networks Topic Guidance
Mathematics Standard 1 Year 12 Networks Topic Guidance
Topic focus ........................................................................................................................... 3
Terminology .......................................................................................................................... 3
Use of technology ................................................................................................................ 3
Background information ...................................................................................................... 3
General comments ............................................................................................................... 4
Future study.......................................................................................................................... 4
Subtopics .............................................................................................................................. 4
MS-N1: Networks and Paths .................................................................................................. 5
Subtopic focus ................................................................................................................. 5
N1.1 Networks ........................................................................................................................ 5
Considerations and teaching strategies .......................................................................... 5
Suggested applications and exemplar questions ............................................................ 6
N1.2 Shortest Paths ............................................................................................................... 7
Considerations and teaching strategies .......................................................................... 7
Suggested applications and exemplar questions .......................................................... 10
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Topic focus
Networks involve the graphical representation and modelling of situations as an approach to
decision-making processes.
Knowledge of networks enables development of a logical sequence of tasks or a clear
understanding of connections between people or items.
Study of networks is important in developing students’ ability to interpret a set of connections
or sequence of tasks as a concise diagram in order to solve related problems.
Terminology
degree of a vertex
directed network ⚑
edge
Kruskal’s algorithm ⚑
map
minimum spanning tree ⚑
network
network diagram
path
Prim’s algorithm ⚑
shortest path ⚑
spanning tree ⚑
tree ⚑
vertex
vertices
weighted edge
Use of technology
Appropriate software could be used in order to construct network diagrams and minimum
spanning trees to represent and analyse networks.
Background information
Very few areas of Mathematics have their history as clearly recorded as Graph Theory and
consequently Networks. The foundations of Graph Theory can be traced officially to 1735
when Euler sought to prove the Königsberg bridge problem. The problem related to a puzzle
the locals pondered: Can a person walk all seven bridges and never cross any given bridge
twice?
It was Euler’s translation of the map of the seven bridges of Königsberg into an abstract
mathematical representation, a graph, using points and lines to represent quantities and
connections to prove the problem that represented the foundations of Graph Theory.
In 1736 Euler published Solutio problematis as geometriam situs pertinentis, which translates
to ‘the solution of a problem relating to the theory of position’ and describes the Königsberg
bridge problem and related proofs. This history is represented in texts related to Graph Theory
or Network theory. Interestingly the term ‘graph’ in this context did not appear until James
Sylvester in the nineteenth century used in relation to diagrams of molecules.
The use of networks is becoming increasingly prolific particularly in representing and analysing
data in areas such as Technological networks, Biological networks, Information networks and
Social and Economic networks.
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General comments
Network theory is sometimes described as graph theory with the use of the word ‘graph’
having a different meaning in this context than that in other mathematics contexts; for instance,
it is not the graph of a function or a data graph.
Other algorithms that are used to determine minimum spanning trees could be explored, for
example Solin’s (Borůvka’s) minimum spanning tree algorithm.
Algorithms that can be used to find the shortest path could be explored, for example Dijkstra’s
algorithm.
Analysing current social media networks using technology is a relevant application of Network
theory and has become a vibrant field of research referred to as Social Network Analysis.
Similar applications of Network theory are found in many other areas, including linguistics,
criminology, medicine and law.
Future study
The study of networks can be continued at a tertiary level and in various vocational contexts,
for example, to examine the efficiency of a delivery service or to minimise costs of connecting
services.
Network theory is becoming increasingly relevant in further study in areas such as Project
Management, Data Analysis and scientific research.
Subtopics
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MS-N1: Networks and Paths
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MS-N1: Networks and Paths
Subtopic focus
The principal focus of this subtopic is to identify and use network terminology and to solve
problems involving networks.
Students develop their awareness of the applicability of networks throughout their lives, for
example social networks and their ability to use associated techniques to optimise practical
problems.
N1.1 Networks
Considerations and teaching strategies
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A network is a collection of objects connected to each other in some way. Networks
are made up of vertices joined by edges.
Different terminology exists to describe aspects of networks including ‘node’ instead of
‘vertex’ and ‘arc’ instead of ‘edge’.
A directed network is a graph in which each edge has a direction, as with a one-way
street. In a directed network the edges have arrows to indicate the direction you may
travel along it.
Networks may or may not have weighted edges, depending on the context.
The weight of an edge may be distances or times, but it could also be something
entirely different, such as depreciation costs or petrol charges.
Edge weights can for instance represent lengths, times, weights, costs, or something
entirely different, for example, a map of a railway system where each station is
represented by a vertex and each edge represents the train lines between two
neighbouring stations. Appropriate edge weights could include travel times between
stations, or costs of running a train between stations.
The terminology should be discussed in detail and students encouraged to use the
appropriate terms. Students could be encouraged to write their own glossary for the
topic of Networks.
Students should be given the opportunity to construct network diagrams from real-life
situations, for example, the Königsberg bridge problem, a simple rail network or an
electricity cable network.
Practical experiments could be conducted and represented by a network diagram,
such as every person at a party shaking hands with every other person at that party
(which can lead to discovery of the Handshake Lemma which states that an even
number of people must have shaken an odd number of other people’s hands). Another
example that could be considered is networks involving personal connections, such as
film actors who are connected (the popular notion “six degrees of separation from
Kevin Bacon”).
It is important to have a way of representing networks which does not rely on a
diagram. In particular, computers cannot work with diagrams. A matrix or table can be
constructed that shows all the information about the weights in a network, such as the
one below:
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𝐴
𝐵
A

B2
C 4

D  

𝐶
𝐷
4 

 3 
3  1

 1  
2
A
B
C
D
A
-
2
4
-
B
2
-
3
-
C
4
3
-
1
D
-
-
1
-
A directed network can also be represented in a table. For example:
From:
To:
A
B
C
D
A
-
2
4
-
B
-
-
3
-
C
5
-
-
1
D
-
-
1
-
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When a network diagram is constructed, reconstructions may differ in appearance. For
example the network diagram below may be constructed from the table above.
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The construction of network diagrams where multiple edges emerge from each vertex
should be explored.
Suggested applications and exemplar questions
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The applications of networks should include both the physical connection of elements
in a network to each other (for example a train network) as well as the appropriate
sequencing of activities within a network to enable its successful completion.
Start and end points should involve genuine activities that students can relate to, for
example, renovating a bathroom; getting ready to go out; obtaining your driver license.
Model the following house plan as a network, showing the doors (doorways) as edges
and the rooms as vertices.
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Given a road atlas or access to an online map, students note the distances and routes
between certain cities in NSW, and draw a network diagram that represents the map.
Investigate network representations of songlines or kinship within the Aboriginal or
Torres Strait Islander cultures.
N1.2 Shortest Paths
Considerations and teaching strategies
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A tree is a connected network, or part of a network that does not contain a cycle. An
example of a tree network is an actual tree, where the vertices are where the trunk and
the branches meet and branch off and the edges are the branch and trunk segments
between vertices. There are no cycles here since branches don’t re-join the trunk or
other branches, at least not usually.
A spanning tree in a network is a tree that contains each vertex.
For a connected graph with 𝑛 vertices, each spanning tree has precisely 𝑛 − 1 edges.
Students should be encouraged to consider and explain why this is true.
A minimal spanning tree in a network is the spanning tree with the total length of its
edges at a minimum. Every vertex is connected to every other vertex by the network,
but the total length of the reduced network is a minimum.
Students could consider how to minimise the cost of a computer wireless network at
their school connecting the main hubs of the school in order to maximise connectivity,
for example considering the following diagram:
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Options for connections that would give a spanning tree include:
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and
Students can then investigate which tree would be a minimal spanning tree for the
network.
Prim’s algorithm is a quick method of finding a minimum spanning tree. The sequence
of steps is as follows to find a minimum spanning tree, 𝑇:
 Select any vertex to be the first vertex of 𝑇.
 Consider the edges which connect vertices in 𝑇 to vertices outside 𝑇.
Pick the one with the minimum weight. Add this edge and the extra vertex to 𝑇.
(If there are two or more edges of minimum weight, choose any one of them).
 Repeat the step above until 𝑇 contains every vertex of the graph.
Students should notice that when they apply Prim’s algorithm, they choose the edge
which is immediately ‘best’ without considering the long-term consequences of their
choice.
Kruskal’s algorithm uses the edge weights directly rather than considering the
connections between vertices. The sequence of steps to find a minimum spanning tree
for a connected graph with 𝑛 vertices is as follows:
 Choose an edge of least weight.
 Choose from those edges remaining, an edge of least weight which does not
form a cycle with already chosen edges. (If there are several such edges,
choose any one of them).
 Repeat the step above until 𝑛 − 1 edges have been chosen (until all vertices
are connected by chosen edges).
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An example of applying Kruskal’s algorithm to a network diagram is as follows:
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There are 6 vertices, so we need 5 edges.
Edge 1: Choose edge BC (least weight)
Edge 2: Choose edge DE (next least weight)
Edge 3: Choose edge AF (could also have chosen CD)
Edge 4: Choose edge CD
Edge 5: Reject edge CE as it creates a loop with CDE
Edge 5: Choose edge CF
5 edges chosen so the minimal spanning tree is complete:
Students can begin investigating a shortest path by considering a network representing
distances between towns and asking how far it is from town 𝐴 to town 𝐽.
A shortest path can also be thought of as a path of minimum weight.
To find a shortest path from 𝐴 to 𝐽 in a network follow this sequence of steps:
 Redraw the network, with circles at each vertex except for 𝐴.
 For all vertices one step away from 𝐴, write down the shortest distance inside a
circle representing the closest vertex.
 For all vertices two steps away from 𝐴, write down the shortest distance from 𝐴
inside each circle representing a vertex.
 Continue this way until 𝐽 is reached.
 The shortest path can then be identified by starting at 𝐽 and moving back to the
vertex from which the minimum value at 𝐽 was obtained, then continuing this
until 𝐴 is reached.
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An example of finding a shortest path is as follows:
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A shortest path is 𝐴𝐵𝐸𝐹𝐽 and has weight 8.
Students should recognise that there are often several shortest paths (of equal length)
between two given vertices.
Suggested applications and exemplar questions
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Students could investigate problems relevant to their local area, such as water supply
to an industrial estate, the local rail network or the minimum length of bitumen road
needed so enough vehicles can visit a number of destinations without getting bogged
in bad weather.
Students could investigate problems involving personal or social networks; for
example, if some friends want to be sure that they keep each other updated about
urgent matters, then they could agree on a network of communication between friends,
possibly including online groups. If they wanted to make sure that everyone was
reliably updated on news and that there wasn’t wasted energy spent on two friends
telling the same friend the news, then the network would have no cycles and would be
a spanning tree. Here, the weights are just 1, and the vertices are each of the friends
and each of the social networks. The edges indicate lines of communication.
(a) Find all the possible spanning trees for the following network:
(b) Which of the spanning trees has minimum weight?
(c) Use Prim’s algorithm, starting with vertex 𝐴, to find the minimum spanning tree.
(d) Show that Kruskal’s algorithm produces the same minimum spanning tree as
Prim’s algorithm.
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Find a minimum spanning tree for the following network:
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Find the length of the shortest path from 𝑎 to 𝑒 in the above network.
This table shows the travelling times in minutes between towns which are connected
directly to each other. Note: The dash in a box indicates that towns are not connected
directly to each other:
A
A
B
D
E
0
50 20 25
-
B 50
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25 30 30
C 20 25
0
-
60
D 25 30
-
0
70
30 60 70
0
E
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0
C
-
(a) Draw a network diagram showing the information on this table.
(b) Find the shortest travelling time between A and E.
As a practical experiment, model a network diagram such as the one above by using
pieces of string tied to small key rings. The shortest paths between two vertices could
then be found by pulling the two key rings tightly apart. The shortest path is shown by
the tight strings. Students can then discuss the limitations of this type of physical model
and compare results with that of using theoretical processes.
The distinction between minimum spanning trees and shortest paths could be
explored. For example, the mayor of a small outback town would like to upgrade the
dirt tracks in the town to paved roads so that every home is connected to every other
home by a paved road. The mayor has two choices: minimise the overall kilometres
required (ie the cost) to find a minimum spanning tree or choose the roads to create a
collection of shortest paths from the mayor’s house to every other home.
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