Year 12 ATAR Mathematics Applications · 2020. 4. 21. · applications

Year 12 ATAR Mathematics Applications

Graphs and Networks

Year 12 | ATAR | Mathematics Applications | Graphs and Networks | © Department of Education WA 2020

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Year 12

Mathematics Applications Graphs and Networks

Except where indicated, this content is © Department of Education Western Australia 2020 and released under a Creative Commons CC BY NC licence. Before re-purposing any third-party content in this resource refer to the owner of that content for permission.

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Year 12 Mathematics Applications Graph and Networks

© Department of Education Western Australia 2020 1

Contents

Signposts… ............................................................................................................................... 3

Overview …. .............................................................................................................................. 4

Lesson 1 – Definitions and terminology ...................................................................... 6

Lesson 2 – Bipartite Graphs and Adjacency Matrices .......................................... 15

Lesson 3 – Planar Graphs and Euler’s Formula ....................................................... 26

Lesson 4 – Paths, trails, walks and cycles ................................................................. 35

Lesson 5 – Eulerian and Hamiltonian Graphs .......................................................... 43

Further practice ................................................................................................................... 51

Glossary/Summary ............................................................................................................ 55

Solutions. ............................................................................................................................... 62



Signposts

Each symbol is a sign to help you.

Here is what each one means.

Important Information

Mark and Correct your work

You write an answer or response

Use your CAS calculator

A point of emphasis

Refer to a text book

Contact your school teacher (if you can)

Check your school about Assessment submission



Overview This Booklet contains approximately 4 weeks, or 16 hours, of work. Some students may require additional time.

To guide the pace at which you work through the booklet refer to the content page.

Space is provided for you to write your solutions in this PDF booklet. If you need more space, then attach a page to the page you are working on.

Answers are given to all questions: it is assumed you will use them responsibly, to maximise your learning. You should check your day to day lesson work.

Assessments

All of your assessments are provided for you separately by your school.

Assessments will be either response or investigative. Weightings for assessments are provided by your school.

Calculator

This course assumes the use of a CAS calculator. Screen displays will appear throughout the booklets to help you with your understanding of the lessons. Further support documents are available.

Textbook

You are encouraged to use a text for this course. A text will further explain some topics and can provide you with extra practice questions.

Online Support

Search for a range of online support.



Content covered in this booklet

The syllabus content focused on in this booklet includes:

The definition of a graph and associated terminology

3.3.1 demonstrate the meanings of, and use, the terms: graph, edge, vertex, loop, degree of a vertex, subgraph, simple graph, complete graph, bipartite graph, directed graph (digraph), arc, weighted graph, and network 3.3.2 identify practical situations that can be represented by a network, and construct such networks 3.3.3 construct an adjacency matrix from a given graph or digraph and use the matrix to solve associated problems Planar graphs demonstrate the meanings of, and use, the terms: planar graph and face 3.3.5 apply Euler’s formula, v+f-e=2 to solve problems relating to planar graphs Paths and cycles 3.3.6 demonstrate the meanings of, and use, the terms: walk, trail, path, closed walk, closed trail, cycle, connected graph, and bridge 3.3.7 investigate and solve practical problems to determine the shortest path between two vertices in a weighted graph (by trial-and-error methods only) 3.3.8 demonstrate the meanings of, and use, the terms: Eulerian graph, Eulerian trail, semi-Eulerian graph, semi-Eulerian trail and the conditions for their existence, and use these concepts to investigate and solve practical problems 3.3.9 demonstrate the meanings of, and use, the terms: Hamiltonian graph and semi-Hamiltonian graph, and use these concepts to investigate and solve practical problems



Lesson 1

The definition of a graph and associated terminology By the end of this lesson you should be able to:

• demonstrate the meanings of, and use, the terms: graph, edge, vertex, loop, degree of a vertex, subgraph, simple graph, complete graph, directed graph (digraph), arc, weighted graph, and network

• identify practical situations that can be represented by a network, and construct such networks

Networks/Graphs

A graph, is a set of points, called vertices (single: vertex), that are connected by lines called edges. By making these connections, we create regions. We will use the terms networks and graphs interchangeably.

This graph (network) has 6 edges

This network has 4 vertices

Vertices can also be called ‘nodes’



There are 4 regions in this network.

Regions can also be called ‘faces’.

The ‘outside’ of the network also counts as a region, see region 4 opposite.

Skills Development 1.1

How many vertices, edges and regions do each of the following have? 1. 2.

3. 4. 5. 6.

You will find more practice exercises in your school textbook or online.



More vocabulary Loop: Any edge that starts and begins at the same node (vertex) is called a loop.

In this graph, the edge that starts and ends at vertex B is a loop. Multiple edges: If more than one edge (or arc) connects the same pair of vertices, these

edges are described as multiple edges.

This graph contains multiple edges because there are 3 different edges that join the nodes B and C.

Complete Graphs: A complete graph is one in which every vertex is connected to every other vertex by a single edge.

Complete graphs are given the notation 𝐾𝐾𝑛𝑛where 𝑛𝑛 is the number of vertices. The complete graphs 𝐾𝐾3,𝐾𝐾4 and 𝐾𝐾5are shown in the following diagram (next page).



Complete graphs are undirected. A complete graph with 𝒏𝒏 vertices will have 𝒏𝒏(𝒏𝒏−𝟏𝟏)

𝟐𝟐 edges

Weighted graph: A weighted graph contains information on its edges representing some quantity or measurement. Examples could be length, cost, flow capacity, etc. It is important to note that weighted graphs are not scale diagrams, the length of an edge does not represent the quantity being displayed.

Here are some examples of weighted graphs:

a. Distances from a point

b. Cost of cabling for a network



c. Capacity, in litres, of pipes in a plumbing system

Directed graphs (digraphs) In Example c. in the previous section, it would be more realistic to assume that the flow in those pipes would follow a particular direction. It would be a pretty inefficient delivery system if pipes were allowing flow in opposite directions. If we drew this graph with arrows showing the direction of flow, then we have created what is known as a directed graph or digraph. A graph with no directed edges is called an undirected graph or non-directed graph. An undirected graph can be transformed into a directed graph by the use of multiple edges, with arrows in opposing directions. However, this practice is rarely used.



You should note that in Figure 2 on the previous page, not every edge has an arrow. In this case, it is assumed that there would be arrows pointing in both directions along that edge. This is the equivalent of a ‘two-way’ street. Simple Networks: A simple network (or simple graph) is a network that is undirected and contains no multiple edges or loops. Subgraphs: A subgraph is a graph that is itself taken from a larger graph. For example:




1. A technology company is having cable installed to connect workstations. The network below represents the cost of cabling each section. Calculate the total cost of the installation.

2. The diagram below shows the distances between towns on a map in kilometres (not to scale).

List the possible paths between Alphaville, at point A and Epsilon Town at point E. Find the shortest path that can be travelled between the two towns. You must always travel in the direction away from Alphaville towards Epsilon Town.



3. Look carefully at each of the networks below. Complete the questions which follow.

a.

b.

c.

d.

e.

f.

g.

h.

i.

j.

k.

l.

List which of the above networks satisfy each of the following: a. Is a simple graph:

b. Has multiple edges:

c. Contains a loop:

d. Is a digraph:

e. Is a digraph with multiple edges:

f. Is a digraph with a loop:



4. In the space provided, sketch the complete graph 𝐾𝐾6.

5. 𝐾𝐾8 is the complete graph with 8 vertices. Without drawing the graph of 𝐾𝐾8, find how many edges it has.




Lesson 2 Bipartite Graphs and Adjacency Matrices

By the end of this lesson you should be able to:

• Demonstrate the meaning of, and use bipartite graphs, • construct an adjacency matrix from a given graph or digraph and use the matrix to solve associated

problems.

The graph below shows which of three events (long jump, javelin, discus) that four athletes compete in.

This type of graph is called a bipartite graph or bigraph. A bipartite graph consists of two groups of vertices where edges join members of one group to the other, but there are no edges between members of the same group.



Another example would be a bipartite graph linking actors and movies:

Note that is not a complete bipartite graph.

A complete bipartite graph has every element in each group connected to every element in the other group. An example of a complete bipartite graph would be:



Adjacency Matrices When two vertices of a graph are joined by an edge, the vertices are said to be adjacent. The connections between the nodes in a network can be represented as an adjacency matrix. Consider the network below:

To construct an adjacency matrix for this network, we draw an 𝑛𝑛 × 𝑛𝑛 matrix, where 𝑛𝑛 is the number of nodes. Let us call the matrix 𝑀𝑀. The entry in cell 𝑚𝑚𝐴𝐴𝐴𝐴 represents the number of connections that join A to B in that direction. The entry in cell 𝑚𝑚𝐴𝐴𝐵𝐵 represents the connections that join B to D in that direction, and so on.

𝑀𝑀 = �

𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐵𝐵𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐵𝐵𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐴𝐴 𝑚𝑚𝐴𝐴𝐵𝐵𝑚𝑚𝐵𝐵𝐴𝐴 𝑚𝑚𝐵𝐵𝐴𝐴 𝑚𝑚𝐵𝐵𝐴𝐴 𝑚𝑚𝐵𝐵𝐵𝐵

�

Completing the matrix for the network above: We have no path that takes A back to A, so the entry in position 𝑚𝑚𝐴𝐴𝐴𝐴 is zero. There are 2 ways to travel from A to B so we put 2 at position 𝑚𝑚𝐴𝐴𝐴𝐴. There is one connection from A towards C so we put 1 in position𝑚𝑚𝐴𝐴𝐴𝐴 and continue until all cells are populated. The connection from D back to D is interesting because it has a loop. It is standard practice to count this as 1 for the purpose of completing adjacency matrices. Therefore 𝑚𝑚𝐵𝐵𝐵𝐵 = 1.



Check the matrix below and see if it matches what you would expect from the given network.

For an

undirected graph, the entries will be symmetrical about the leading diagonal.

Graphs with no loops will have a leading diagonal consisting of all zeros.

Example

Complete the adjacency matrix for the graph shown

Notice that this graph is undirected, therefore it will be symmetrical about its leading diagonal. Since there are no loops the leading diagonal will be all zeroes.

Working from A to each of the other nodes we see firstly, there is one path from A to B, therefore 𝑚𝑚𝐴𝐴𝐴𝐴 = 1. Next, there is no path from A to C, so 𝑚𝑚𝐴𝐴𝐴𝐴 = 0. There is one path to D and no path to E, therefore the numbers 1 and 0 complete this row.

Complete the matrix yourself before checking against the solution below.



Well done if you agree that the completed matrix would be:

Let us have another look at the network that produced this matrix:

Note that even though there is no direct link from A to C, it is possible to travel from A to C using two edges. In fact, there are three different paths we could take that use exactly two edges.

Similarly, it is possible to travel from E to A using 2 edges, but only along one path.

Consider the square of the adjacency matrix for this network:

This matrix, 𝑀𝑀2, tells us how many ways there are to travel between each pair of nodes, using exactly 2 edges.

The 2 in position 𝑚𝑚𝐴𝐴𝐴𝐴 indicates that it is possible to travel from node A back to itself using two edges in two different ways. These are A to B and back to A and A to C and back.

The 0 in position 𝑚𝑚𝐴𝐴𝐴𝐴 suggests there is no path of length two that starts at A and ends at B.

As an exercise, see if you can find all six journeys of length two that join C back to itself, as indicated in the matrix.

In general, to find the number of possible paths of length 𝑛𝑛 there are between the nodes of a network, the adjacency matrix is raised to the power of 𝑛𝑛.



If 𝑀𝑀 is the adjacency matrix then

𝑀𝑀2 gives the number of paths of length 2 between nodes

𝑀𝑀3 gives the number of paths of length 3 between nodes

In general:

𝑀𝑀𝑛𝑛 gives the number of paths of length 𝑛𝑛 between each pair of nodes.


1. There are four machines on a factory floor. Three workers are employed at the factory and are trained to work on different machines. Employee A is trained to use machine 1 and machine 4 Employee B is trained to use machines 2, 3 and 4 Employee C is trained to use machines 2 and 3. Draw a bipartite graph to represent this information.



2. Some AFL players have played for more than one club during their careers. Some of the players are listed here, along with the clubs they played for

Koby Stevens West Coast/Western Bulldogs/St. Kilda Stewart Crameri Essendon/Western Bulldogs/Geelong Joel Hamling Geelong/Western Bulldogs/Fremantle Show this information in a bipartite graph

3. The following graph shows the bus connections between some towns.

Draw an adjacency matrix to represent this information.



4. a. Construct an adjacency matrix to represent the information given in the following

network:

b. By raising this matrix to a suitable power, list all of the nodes that can be reached from 𝐴𝐴 in exactly 2 steps.



5. The matrix A is an adjacency matrix.

Without drawing this graph, write down: a. How you know the graph represented by this matrix is undirected b. How you know this graph contains no loops.

6. Construct the adjacency matrix that would represent this network.



7. From the adjacency matrix provided below, sketch a possible network that it could represent.

8. The adjacency matrix for a particular graph does not have a leading diagonal consisting of all zeros. It is also not symmetrical about the leading diagonal.

Write down two properties of the graph represented by this adjacency matrix.

9. You are told that a graph has no edges that cross. You are given its adjacency matrix. Write down the properties would you look for in the adjacency matrix if you were checking whether or not the graph is a simple graph.



10. 𝐴𝐴 is an adjacency matrix representing the traffic system in a small town. The digits represent the number of streets that directly link each pair of nodes on a graph drawn of the system.

a. How can you tell from this matrix there are no one-way streets in this town? b. Find 𝐴𝐴2

c. Hence, write down how many different ways there are to travel from point C to A using

exactly 2 streets.




Lesson 3

Planar Graphs and Euler’s Formula


• demonstrate the meanings of, and use, the terms: planar graph and face • apply Euler’s formula, 𝑣𝑣 + 𝑓𝑓 − 𝑒𝑒 = 2 to solve problems relating to planar graphs

Planar graphs: A graph is planar if, and only if, it can be drawn without any edges crossing.

A graph may not initially look planar, but if an edge can be redrawn so it does not cross any other edges, it is planar.

For example, this graph has intersecting edges:

however, it can be redrawn as shown:

Since there are no edges crossing, this graph is planar.



The graph above is the complete graph 𝐾𝐾4 which happens to be the largest complete graph that is planar.

This means it is not possible to draw a complete graph with more than 4 vertices without crossing edges.

Connected graphs: A graph is said to be connected if it is possible to travel from any vertex to another vertex along edges of the graph. In figure 1, even though you have to travel along multiple edges, it is possible to get to any node from any other node. This makes it a connected graph.

In figure 2, there is no possible path from 𝑀𝑀 to 𝑃𝑃, therefore this is a disconnected graph or unconnected graph.




1. Redraw this graph so that none of its edges cross (is planar).



2. Redraw this graph to show that is planar.

3. Can the following graph be redrawn without any edges intersecting? Why/Why not?



Euler’s Formula Recall from Lesson 1 that when nodes are joined by edges, they create regions or faces. We are going to use the term ‘faces’ for this part of our topic. Reminder:

This graph has 3 vertices, 4 edges and 3 faces, numbered. It is important to remember that we consider the ‘outside’ a face. In this case, 1 and 2 are known as finite faces and 3 is an infinite face. Consider the following planar graphs. In the space provided is written the number of vertices (𝑣𝑣), faces (𝑓𝑓) and edges (𝑒𝑒) in each graph.

𝑣𝑣 = 4 𝑓𝑓 = 3 𝑒𝑒 = 5

𝑣𝑣 = 3 𝑓𝑓 = 3 𝑒𝑒 = 4

𝑣𝑣 =5 𝑓𝑓 = 4 𝑒𝑒 = 7

𝑣𝑣 = 6 𝑓𝑓 = 5 𝑒𝑒 =9

𝑣𝑣 =8 𝑓𝑓 = 6 𝑒𝑒 =12

𝑣𝑣 =4 𝑓𝑓 =4 𝑒𝑒 = 6



Take a moment to see if you can spot a relationship between 𝑣𝑣,𝑓𝑓 and 𝑒𝑒 that holds true for all of the above graphs. There is a relationship between 𝑣𝑣,𝑓𝑓 and 𝑒𝑒, called Euler’s Formula, named for Leonhard Euler (1707-1783) who was a very famous Swiss mathematician and astronomer. His surname is pronounced ‘oiler’. Well done if you spotted that if you find 𝑣𝑣 + 𝑓𝑓 − 𝑒𝑒, the answer is 2 for all of the above graphs. In general, we can say: For any planar graph with 𝑣𝑣 vertices, 𝑓𝑓 faces and 𝑒𝑒 edges: 𝒗𝒗 + 𝒇𝒇 − 𝒆𝒆 = 𝟐𝟐 Example 1: A planar graph has 6 vertices and 7 edges.

How many faces does this graph have?

Solution 1:

Using Euler’s rule we have 𝑣𝑣 + 𝑓𝑓 − 𝑒𝑒 = 2

We have 𝑣𝑣 = 6 and 𝑒𝑒 = 7 and wish to find 𝑓𝑓.

Substituting the values given we get

6 + 𝑓𝑓 − 7 = 2

which gives 𝑓𝑓 = 3 when solved.

The graph contains 3 faces.

A possible graph would be:




1. A connected planar graph has 5 vertices and 7 edges, how many faces will it have?

2. a. How many edges does a planar graph if it contains 4 vertices and 5 faces?

b. Draw a sketch of a possible network with these properties.

3. Is a graph that contains 6 vertices, 3 faces and 8 edges a planar graph? Justify your decision with reasoning.



4. A sketch of a rectangular prism is given below. By counting vertices, edges and faces, show that Euler’s Formula applies to this prism.

5. Show that Euler’s Formula applies to a triangular prism. (A sketch will help).



6. Draw two subgraphs of the graph drawn below.

7. A graph is drawn below.

a. Draw a subgraph of this graph. b. Draw a subgraph of your answer from a.




Lesson 4 By the end of this lesson you should be able to:

• demonstrate the meanings of, and use, the terms: walk, trail, path, closed walk, closed trail, cycle, connected graph, and bridge

• investigate and solve practical problems to determine the shortest path between two vertices in a weighted graph (by trial-and-error methods only)

Paths, trails and walks

More definitions

As we travel through a network, we need some vocabulary to properly describe the different ways we can do this.

The graph below will be used to describe the different definitions

Walk: A walk is a sequence of vertices, with edges between them from each vertex to the next. For example, in Figure 1, a walk could be from 𝐴𝐴 to 𝐵𝐵 along the route that goes from 𝐴𝐴 to 𝐷𝐷 to 𝐵𝐵. This would be written as 𝐴𝐴𝐷𝐷𝐵𝐵. Another walk from 𝐴𝐴 to 𝐵𝐵 could be 𝐴𝐴𝐴𝐴𝐵𝐵. Another walk could be 𝐴𝐴𝐷𝐷𝐴𝐴𝐵𝐵𝐷𝐷𝐴𝐴.

If a walk starts and ends at two different vertices it is called an open walk. If a walk starts and finishes at the same vertex, it is a closed walk.

In Figure 1, 𝐴𝐴𝐴𝐴𝐵𝐵 is an open walk, while 𝐴𝐴𝐷𝐷𝐴𝐴𝐴𝐴 is a closed walk.

Path: A path is a walk with special conditions attached. If a walk has no repeated edges and no repeated vertices (apart from the last one), the walk is a path. In the network above, 𝐴𝐴𝐷𝐷𝐴𝐴𝐵𝐵 is a path. As it starts and finishes at different vertices, it is an open path. 𝐴𝐴𝐴𝐴𝐵𝐵𝐷𝐷𝐴𝐴 is a closed path as it starts and finishes at the same vertex.



Trail: A trail is a walk that has no repeated edges. In Figure 1, 𝐴𝐴𝐴𝐴𝐵𝐵𝐷𝐷𝐴𝐴 is a trail. As expected, an open trail starts and finishes at different vertices, while closed trails start and finish at the same vertex.

All paths are trails (no repeated use of edges or vertices) but not all trails are paths (there may be some use of repeated vertices).

In Figure 2 there is a walk from 𝐴𝐴 to 𝐸𝐸, 𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵𝐸𝐸.

𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵𝐸𝐸 is a trail because there is no repeated use of any edge. However, it is not a path as the vertex 𝐵𝐵 is passed through twice. It is only acceptable to pass through the same vertex twice if it is at the start and finish of the walk.

𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵𝐸𝐸 is an open trail. 𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵 is a closed path.

Cycle: A cycle is a closed walk which begins and ends at the same vertex and which has no repeated edges or vertices except the first. In the previous example, 𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵 is a cycle.

Length: The length of a walk, trail or cycle is the number of edges it uses. In Figure 2 above, the walk 𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵𝐸𝐸 is of length 5 as it travels along 5 edges. The cycle 𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵 has length 3.

Bridge: A bridge is an edge in a connected graph (see Lesson 3) that, if removed would cause the graph to become disconnected. In Figure 2, the removal of either edge 𝐴𝐴𝐵𝐵 or 𝐵𝐵𝐸𝐸 would cause the graph to become disconnected. 𝐴𝐴𝐵𝐵 and 𝐵𝐵𝐸𝐸 are bridges. However, if we removed any of edge 𝐵𝐵𝐴𝐴, 𝐴𝐴𝐷𝐷 or 𝐵𝐵𝐷𝐷, the graph would remain connected. These are not bridges.




1. From the graph on the right:

a. Identify a walk of length 3 b. Write down a walk of length 4 from A to B c. Identify a cycle of length 3:

d. Identify a path from A to B of length 3.

e. Describe 𝐴𝐴𝐷𝐷𝐵𝐵𝐴𝐴𝐴𝐴 using correct terminology.

f. Explain why 𝐴𝐴𝐷𝐷𝐴𝐴𝐴𝐴𝐵𝐵 is a trail but not a path.

g. Are the any bridges in this graph? Why/why not?



2. From the graph at right, write down: a. A walk of length 4 b. A path of length 5 c. Two cycles d. A trail of length 5 e. A bridge f. A description of 𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝐷𝐷 using correct terminology g. An explanation of why 𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝐷𝐷𝐸𝐸 is a trail but not a path.




Practical problems involving shortest paths In Unit 4 of Mathematics Applications you will study some formal methods for identifying the shortest path between two vertices in a weighted network or graph. In Unit 3, the approach is a trial-and-error one, where we calculate all of the possible lengths from the graph and choose the optimum from our answers. I will do some worked examples of how to answer questions in this way. Example 1 Figure 1 represents an obstacle course (not to scale). Each node in the graph represents a task to be completed. Participants must complete 3 different tasks as well as make it from the starting point at 𝐴𝐴 to the finishing line at 𝑍𝑍. All of the tasks usually take the same time to complete.

Find the shortest path between 𝐴𝐴 and 𝑍𝑍 that passes through 3 tasks and write down this length.

Solution 1: Let us write down each possible path and calculate its length

𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝑍𝑍: 220 + 310 + 280 + 270 = 1080 𝑚𝑚

𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝑍𝑍: 220 + 310 + 230 + 490 = 1250 𝑚𝑚

𝐴𝐴𝐴𝐴𝐴𝐴𝐷𝐷𝑍𝑍: 310 + 300 + 320 + 850 = 1780 𝑚𝑚

𝐴𝐴𝐴𝐴𝐷𝐷𝐴𝐴𝑍𝑍: 310 + 220 + 320 + 490 = 1340 𝑚𝑚

𝐴𝐴𝐷𝐷𝐴𝐴𝐴𝐴𝑍𝑍: 120 + 220 + 300 + 490 = 1130 𝑚𝑚

The shortest path is along 𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝑍𝑍 and is 1080 𝑚𝑚 in length.



Example 2 Gerald delivers for Uber Eats. He has room to carry four deliveries in his car. He picks up four meals and sets out on his final delivery route of the day. He leaves Burn’t at point 𝐴𝐴 and wants to deliver the meals before heading directly home to point 𝐻𝐻 after the final delivery. Figure 2 shows the positions of the houses to which Gerald will deliver and the travel time between them, in minutes, on a weighted graph (not to scale). Gerald does not wish to travel along the same street twice. What is the shortest route he can take to deliver the four meals and continue to home?

Solution 2: This an undirected graph, meaning there is no particular direction that we

must travel along any edge (the streets are all two-way).

𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐸𝐸𝐻𝐻: 12 + 4 + 5 + 1 + 14 = 36 𝐴𝐴𝐵𝐵𝐷𝐷𝐴𝐴𝐸𝐸𝐻𝐻: 12 + 2 + 5 + 3 + 14 = 36

𝐴𝐴𝐵𝐵𝐴𝐴𝐸𝐸𝐷𝐷𝐻𝐻: 12 + 4 + 3 + 1 + 12 = 32 𝐴𝐴𝐷𝐷𝐵𝐵𝐴𝐴𝐸𝐸𝐻𝐻: 8 + 2 + 4 + 3 + 14 = 31 𝐴𝐴𝐷𝐷𝐴𝐴𝐵𝐵𝐷𝐷𝐸𝐸𝐻𝐻: = 8 + 5 + 4 + 2 + 1 + 14 = 34 𝐴𝐴𝐴𝐴𝐵𝐵𝐷𝐷𝐸𝐸𝐻𝐻: 7 + 4 + 2 + 1 + 14 = 28 𝐴𝐴𝐴𝐴𝐷𝐷𝐵𝐵𝐴𝐴𝐸𝐸𝐻𝐻: 7 + 5 + 2 + 4 + 3 + 14 = 35 The shortest time that Gerald can take between 𝐴𝐴 and 𝐻𝐻 is 28 𝑚𝑚𝑚𝑚𝑛𝑛𝑚𝑚𝑚𝑚𝑒𝑒𝑚𝑚 and is along the route 𝐴𝐴𝐴𝐴𝐵𝐵𝐷𝐷𝐸𝐸𝐻𝐻




1. Find the shortest way to get from point 𝐴𝐴 to point 𝑋𝑋 in the following graph (not to

scale), passing through all other nodes on the way:

2. In the following graph, find the shortest path from 𝑃𝑃 to 𝑍𝑍, passing through all nodes at

least once.



3. A delivery driver wishes to minimise fuel and is planning the shortest route he can take to deliver his goods. He does not allow himself to travel along the same street twice (in any direction). The delivery drop-off points are shown on the network below. a) Find the shortest route he can take to deliver to every drop-off point and write down the distance travelled. b) A new road of length 1km is built between B and D. What is now the shortest

route? How long is this route?

You will find more practice questions in your textbook or online.



Lesson 5


• demonstrate the meanings of, and use, the terms: Eulerian graph, Eulerian trail, semi-Eulerian graph, semi-Eulerian trail and the conditions for their existence, and use these concepts to investigate and solve practical problems.

Traversability and the order of a vertex A popular puzzle is to attempt to draw the following shape without lifting your pen from the page or travelling along the same edge twice. It is permissible to pass through a vertex more than once:

It is possible, but you will notice that your correct attempts either started at the bottom left corner and finished at the right bottom corner or vice-versa. This has to do with a property of the vertices in the graph, known as their degree or order. In an undirected graph, the order (or degree) of a vertex is the number of edges that meet at this vertex. In a directed graph, we describe two different degrees: the in-degree which is the number of edges pointing towards that vertex and the out-degree which is the number of edges pointing away from that vertex. Revisiting our puzzle, we can write down the order of each vertex. For ease of reference, the vertices will be labelled 𝐴𝐴 to 𝐸𝐸.



We write 𝑑𝑑𝑒𝑒𝑑𝑑(𝐴𝐴) = 3, deg(𝐵𝐵) = 4, deg(𝐴𝐴) = 2, etc. 𝐴𝐴 and 𝐸𝐸 are called ‘odd’ matrices because their order is an odd number. 𝐵𝐵,𝐴𝐴 and 𝐷𝐷 are ‘even’ vertices. As we are able to draw the shape above without raising our pen or travelling along an edge twice, it is said to be traversable. Consider some more traversable graphs:

Take a moment to study these graphs; write down the degree of each vertex and see if you can spot a pattern before looking at the answer.

Vertex Order A 3 B 4 C 2 D 4 E 3



Well done if you noticed that all of the graphs above fulfilled either one of two conditions. A graph is traversable if it is connected, and

• it has all vertices of even degree. In this case, any point can be the starting and finishing point

• it has exactly 2 vertices of odd degree. In this case one odd vertex is the starting point and the other odd vertex is the finishing point, i.e.

A graph is traversable if has exactly 𝟎𝟎 or 𝟐𝟐 odd vertices. Eulerian and Semi-Eulerian Graphs Leonhard Euler (1707-1783) solved the famous ‘Seven Bridges of Konigsberg’ problem https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg, which he reduced to a graph theory problem. As a result, his name is connected to certain types of trails and graphs: A graph is considered to be Eulerian if it is connected and has a closed trail (a walk with no repeated edges) containing all edges of the graph. Essentially, a graph is considered Eulerian if you can start at a vertex, traverse through every edge only once, and return to the same vertex you started at. We saw earlier that is only possible when there are no odd vertices.

A Eulerian Graph has vertices of only even degree.

This graph contains vertices of degree 2 or 6 and no odd vertices. It is a Eulerian Graph. You can start a closed trail at any vertex.

https://en.wikipedia.org/wiki/Seven_Bridges_of_K%C3%B6nigsberg



A graph is considered Semi-Eulerian if it is connected and it has an open trail containing every edge of the graph (exactly once as per the definition of a trail). You do not need to return to the starting vertex. We saw earlier that this happens when you have 2 odd vertices.

A semi-Eulerian graph has exactly 2 odd vertices.

There are 2 odd vertices in this graph, 𝐴𝐴 and 𝐵𝐵. You can follow an open trail that starts at 𝐴𝐴 and ends at 𝐵𝐵 or starts at 𝐵𝐵 and ends at 𝐴𝐴. Hamiltonian and Semi-Hamiltonian graphs William Hamilton (1805-1865) was an Irish mathematician who, amongst many discoveries, studied closed paths along the edges of the dodecahedron (a solid with 12 faces, all pentagonal) that visit each vertex only once. Such paths became known as Hamiltonian circuits. Whereas Euler was concerned with travelling along edges only once, Hamilton was concerned with visiting vertices only once. A Hamiltonian path visits every vertex in a graph once only, with the exception of possibly starting and finishing at the same vertex.

If the path begins and ends at the same vertex, i.e. a closed path, it is called a Hamiltonian cycle. A connected graph that contains a Hamiltonian cycle is called a Hamiltonian graph. A connected graph that contains a Hamiltonian path that is not a cycle (i.e. an open path), is called a semi-Hamiltonian graph. There is no simple rule for checking if a graph is Hamiltonian or semi-Hamiltonian.



Example 1: Other than 𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐸𝐸𝐴𝐴, write down a Hamiltonian path from the graph provided:

Is this a Hamiltonian or semi-Hamiltonian graph? Justify your answer. Solution 1: There are a number of Hamiltonian paths, one of which is 𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐸𝐸𝐷𝐷

This is a semi-Hamiltonian graph because the path is open, it starts and finishes at different vertices.



Skills development 5.1

1. From the graph provided, write down the degree of each vertex

2. Fin the in-degree and out-degree for each vertex in the following network:

3. Explain, with reasoning, whether or not the graph below is traversable.

Vertex Degree A B C D E

Vertex In-degree Out-degree P Q R



4. The following graph is traversable. Describe a trail, using the order of the vertices travelled, that traverses the graph.

5. Describe each of the following graphs as Eulerian or semi-Eulerian a.

b.

c.



6. The graph below can be traversed.

a) Write down the sequence of vertices for a trail that traverses this graph. b) Is this a Eulerian or semi-Eulerian trail? Explain your answer.

7. Draw a Hamiltonian circuit on the graph below:

Is it possible to draw a Eulerian circuit on this graph? Justify your answer.



Further Practice: Examination question practice

By doing further practice you should be able to:

• Apply understanding of arithmetic and geometric sequences to exam type questions

Examination Type Questions

ATAR 2019 Calculator – Free Examination

Copyright © School Curriculum and Standards Authority 2019 https://senior-secondary.scsa.wa.edu.au/further-resources/past-atar-course-exams/mathematics-past-atar-course-exams

Question 1 (6 marks)

The graph shown represents three buildings A, B and C, with connecting walkways, at a local school.

(a) Why is the graph planar? (1 mark)

(b) Show that the graph satisfies Euler’s formula. (2 marks)

(c) Construct the adjacency matrix for the graph. (3 marks)

A student wishes to carry out closed walks of length two from Building A.

(d) List all his possible walks. (2 marks)

https://senior-secondary.scsa.wa.edu.au/further-resources/past-atar-course-exams/mathematics-past-atar-course-exams




ATAR 2019 Calculator-Free Examination

Copyright © School Curriculum and Standards Authority 2019 https://senior-secondary.scsa.wa.edu.au/further-resources/past-atar-course-exams/mathematics-past-atar-course-exams

Question 4 Parts (a) & (b)

A marine park has attractions with paths connecting them. The vertices on the graph represent the attractions and the numbers on the edges represent the path distances (km) between the attractions. Visitors can either walk around the park or take one of the many shuttle buses that run between attractions.

The manager of the marine park leaves his office, which is located at the entrance/exit (E) and walks to attraction V.

(a) (i) Determine the shortest distance from E to V. (1 mark)

(ii) If the manager needs to pick up some tools left at U on the way, determine the route he should take and the corresponding distance, given he wants to take the shortest route from E to V. (2 marks)

Rachel arrives at the entrance. She wants to complete a Hamiltonian cycle. (b) State the route she should take. (2 marks)





ATAR 2019 Calculator Assumed Examination Copyright © School Curriculum and Standards Authority 2019 https://senior-secondary.scsa.wa.edu.au/further-resources/past-atar-course-exams/mathematics-past-atar-course-exams

Question 12 (6 marks)

Jake, a park ranger, is giving a presentation at a National Park and Wildlife Conference

on possible designs for a new park. Unfortunately, Jake made mathematical errors in his presentation about the paths (represented by edges) and shelter huts (represented by vertices) in the park.

(a) For each of the following statements, the graph drawn by Jake was incorrect. Redraw the graph to match the statement correctly.

(i) This park plan has been drawn as a connected planar graph containing six vertices. (2 marks)

(ii) This park plan has been drawn as a bipartite graph. (3 marks)

Jake also makes the following incorrect statement in his presentation. ‘A park plan can be a complete graph with 21 paths and six shelter huts’.

(b) If the plan must be a complete graph with 21 paths, how many shelter huts should Jake have quoted? (1 mark)





More past examination papers can be found at https://senior-secondary.scsa.wa.edu.au/syllabus-and-support-materials/mathematics/mathematics-applications

You should attempt, and self-mark, as many practice examination questions as you can. The website above contains past examinations and marking keys going back to 2016.

https://senior-secondary.scsa.wa.edu.au/syllabus-and-support-materials/mathematics/mathematics-applications





Summary/Glossary

With the permission of the School Curriculum and Standards Authority, adapted from https://senior-secondary.scsa.wa.edu.au/__data/assets/pdf_file/0004/581260/Mathematics-Applications-Y12-Syllabus-AC-ATAR-2020-GD.pdf

Graphs and networks

Adjacency matrix An adjacency matrix for a non-directed graph with 𝑛𝑛 × 𝑛𝑛 vertices is a 𝑛𝑛 × 𝑛𝑛 matrix in which the entry in row 𝑚𝑚 and column 𝑗𝑗 is the number of edges joining the vertices 𝑚𝑚 and 𝑗𝑗. In an adjacency matrix, a loop is counted as 1 edge.

Example:

Adjacency matrix continued

For a directed graph the entry in row 𝑚𝑚 and column 𝑗𝑗 is the number of directed edges (arcs) joining the vertex 𝑚𝑚 and 𝑗𝑗 in the direction 𝑚𝑚 to 𝑗𝑗. Example:

Bipartite graph A bipartite graph is a graph whose set of vertices can be split into two distinct groups in such a way that each edge of the graph joins a vertex in the first group to a vertex in the second group.

Example:

https://senior-secondary.scsa.wa.edu.au/__data/assets/pdf_file/0004/581260/Mathematics-Applications-Y12-Syllabus-AC-ATAR-2020-GD.pdf

https://senior-secondary.scsa.wa.edu.au/__data/assets/pdf_file/0004/581260/Mathematics-Applications-Y12-Syllabus-AC-ATAR-2020-GD.pdf



Bridge See Connected graph.

Closed path See Path

Closed trail See Trail.

Closed walk See Walk.

Complete graph A complete graph is a simple graph in which every vertex is joined to every other vertex by an edge. The complete graph with 𝑛𝑛 × 𝑛𝑛 vertices is denoted 𝐾𝐾𝑛𝑛. A complete bipartite graph is a bipartite graph where every vertex of the first set is connected to every vertex of the second set.

Connected graph A graph is connected if there is a path between each pair of vertices. A bridge is an edge in a connected graph that, if removed, leaves a graph disconnected.

Cycle A cycle is a closed walk which begins and ends at the same vertex and which has no repeated edges or vertices except the first. If 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 and 𝑑𝑑 are the vertices of a graph, the closed walk 𝑏𝑏𝑐𝑐𝑑𝑑𝑏𝑏 that starts and ends at vertex 𝑏𝑏 (shown dotted) an example of a cycle.

Degree of a vertex (graph)

In a graph, the degree of a vertex is the number of edges incident with the vertex, with loops counted twice. It is denoted deg 𝑣𝑣.

In the graph below, deg 𝑎𝑎 = 4, deg 𝑏𝑏 = 2, deg 𝑐𝑐 = 4 and deg 𝑑𝑑 =2.



Directed graph/ Digraph

A directed graph is a diagram comprising points, called vertices, joined by directed lines called arcs. The directed graphs are commonly called digraphs.

Edge See Graph.

Euler’s formula For a connected planar graph, Euler’s rule states that 𝑣𝑣 + 𝑓𝑓 − 𝑒𝑒 = 2, where 𝑣𝑣 is the number of vertices, 𝑒𝑒 is the number of edges and 𝑓𝑓 is the number of faces.

Eulerian graph A connected graph is Eulerian if it has a closed trail (starts and ends at the same vertex), that is, includes every edge and once only; such a trail is called an Eulerian trail. An Eulerian trail may include repeated vertices. A connected graph is semi-Eulerian if there is an open trail that includes every edge once only.

Face The faces of a planar graph are the regions bounded by the edges, including the outer infinitely large region. The planar graph shown has four faces.

Food web A food web (or food chain) depicts feeding connections (who eats whom) in an ecological community.

Graph A graph is a diagram that consists of a set of points, called vertices, that are joined by a set of lines called edges. Each edge joins two vertices. A loop is an edge in a graph that joins a vertex in a graph to itself. Two vertices are adjacent if they are joined by an edge. Two or more edges which connect the same vertices are called multiple edges.



Hamiltonian cycle

A Hamiltonian path is a path that includes every vertex in a graph once only. A Hamilton path that begins and ends at the same vertex is a Hamiltonian cycle. These concepts are useful in solving practical problems, such as: planning a sight-seeing tourist route around a city, or the travelling-salesman problem.

Königsberg bridge problem

The Königsberg bridge problem asks: Can the seven bridges of the city of Königsberg all be traversed in a single trip that starts and finishes at the same place?

Length (of a walk)

The length of a walk is the number of edges it includes.

Multiple edges

See Graph.

Network The word network is frequently used in everyday life, for example, television network, rail network. Practical situations that can be represented by the construction of a network include: trails connecting camp sites in a National Park, a social network, a transport network with one-way streets, a food web, theresults of a round-robin sporting competition.

Weighted graphs or digraphs can often be used to model such networks.

Open path See Path.

Open walk See Walk.

Open trail

See Trail.

Path (in a graph) A path in a graph is a walk in which all of the edges and all the vertices are different. A path that starts and finishes at different vertices is said to be open, while a path that starts and finishes at the same vertex is said to be closed. A cycle is a closed path.

If 𝑎𝑎 and 𝑑𝑑 are the vertices of a graph, a walk from 𝑎𝑎 to 𝑑𝑑 along the dotted edges is a path. Depending on the graph, there may be multiple paths between the same two vertices, as is the case here.



Planar graph A planar graph is a graph that can be drawn in the plane. A planar graph can always be drawn so that no two edges cross.

Semi-Eulerian graph

See Eulerian graph.

Simple graph A simple graph has no loops or multiple edges.

Subgraph

When the vertices and edges of a graph 𝐴𝐴 (shown dotted) are also vertices and edges of the graph 𝐴𝐴, graph 𝐴𝐴 is said to be a subgraph of graph 𝐴𝐴.

Trail A trail is a walk in which no edge is repeated.

The travelling salesman problem

The travelling salesman problem can be described as follows: Given a list of cities and the distance between each city, find the shortest possible route that visits each city exactly once. While in simple cases this problem can be solved by systematic identification and testing of possible solutions, there is no known efficient method for solving this problem.

Vertex See Graph.

Walk (in a graph)

A walk in a graph is a sequence of vertices such that from each of its vertices there

is an edge to the next vertex in the sequence. A walk that starts and finishes at

same vertex is said to be closed walk.



If 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 and 𝑑𝑑 are the vertices of a graph with edges 𝑎𝑎𝑏𝑏, 𝑏𝑏𝑐𝑐, 𝑐𝑐𝑐𝑐, 𝑐𝑐𝑑𝑑 and 𝑏𝑏𝑑𝑑, then the sequence of edges (𝑎𝑎𝑏𝑏, 𝑏𝑏𝑐𝑐, 𝑐𝑐𝑐𝑐, 𝑐𝑐𝑑𝑑) constitute a walk. The route followed on this walk is shown dotted on the graph below.

This walk is denoted by the sequence of vertices 𝑎𝑎𝑏𝑏𝑐𝑐𝑐𝑐𝑑𝑑. The walk is open because

it begins and finishes at different vertices.

A walk can include repeated vertices (as is the case above) or repeated edges.

A example of a closed walk with both repeated edges, and hence vertices, is

defined by the sequence of edges (𝑎𝑎𝑏𝑏, 𝑏𝑏𝑑𝑑, 𝑑𝑑𝑏𝑏, 𝑏𝑏𝑎𝑎) and is denoted by the sequence

of vertices 𝑎𝑎𝑏𝑏𝑑𝑑𝑏𝑏𝑎𝑎. The route followed is shown dotted in the graph below.

Depending on the graph, there may be multiple walks between the same two vertices, as is the case here.

Weighted graph A weighted graph is a graph in which each edge is labelled with a number used to represent some quantity associated with the edge. For example, if the vertices represent towns, the weights on the edges may represent the distances in kilometres between the towns.

Copyright © School Curriculum and Standards Authority 2019



Solutions Skills Development 1.1

1. v=4, e=3, r (or f) =1

2. v=3, e=4, r(or f)=3

3. v=4, e=5, r = 3

4. v=5, e=6, r=5

5. v=4, e=5, r=4

6. v=4, e=7, r=5


1. 220 + 110 + 310 + 110 + 95 + 80 + 120 = 1045, 𝑚𝑚𝑒𝑒 $1045

2.

ABDE: 27.8 km ABCDE: 23.4 km ABCE: 25.1 km ACE: 28.9 km AE:32 km

The shortest path is ABCDE at 23.4 km

3. a) Simple graphs are: a, c, g

b) Multiple edges: b, f, j, k

c) Loops are in: d, l (k is not a loop, look carefully)

d) Digraphs (have arrows): d, e, i, j, l

e) Digraphs & multiple edges: j

f) Digraph & loop: d, l

4.



5. No. of edges in a complete graph is 𝑛𝑛(𝑛𝑛−1)2

8×72

= 28 edges


1.

2.

3. 𝐴𝐴 = �

0 1 1 10 0 1 01 1 0 01 0 1 0

�

4. a) 𝐴𝐴 =

⎣⎢⎢⎢⎡0 1 0 1 00 0 2 0 00 1 0 0 01 0 1 0 10 0 1 0 0⎦

⎥⎥⎥⎤ b) 𝐴𝐴2 =

⎣⎢⎢⎢⎡1 0 3 0 10 2 0 0 00 0 2 0 00 2 1 1 00 1 0 0 0⎦

⎥⎥⎥⎤

A, C, E can all be reached in a 2-step journey (the 3 in position 𝑎𝑎13 means there are 3

two-step journeys between A and C). 5. a) The graph is symmetrical on either side of the leading diagonal b) The leading diagonal is all zeros.



6. 𝐴𝐴 =

⎣⎢⎢⎢⎡0 1 0 0 11 1 1 0 00 1 0 0 00 1 1 0 00 0 0 1 0⎦

⎥⎥⎥⎤

7. 8. There is at least one loop in the graph. The graph is directed. 9. Leading diagonal is all zeros (no loops) Matrix has no number bigger than 1 (no multiple edges) The matrix is symmetrical about the leading diagonal (is undirected) 10. a) The matrix is symmetrical about the leading diagonal (undirected graph)

b) 𝐴𝐴2 = �

9 1 4 11 5 2 54 2 3 21 5 2 5

�

c) There are 4 ways to travel between C and A using exactly 2 streets. Skills development 3.1 1.

2. 3. It is the complete graph 𝐾𝐾5. Only complete graphs up to 𝐾𝐾4 are planar.




1. 𝑣𝑣 + 𝑓𝑓 − 𝑒𝑒 = 2 gives f=4

2. a) 4 + 5 − 𝑒𝑒 = 2, gives 𝑒𝑒 = 7

b)

3. 𝑣𝑣 + 𝑓𝑓 − 𝑒𝑒 = 2 for a planar graph.

6 + 3 − 8 = 1Not planar

4. 𝑣𝑣 = 8, 𝑒𝑒 = 12,𝑓𝑓 = 6 Is 𝑣𝑣 + 𝑓𝑓 − 𝑒𝑒 = 2?

8 + 6 − 12 = 2, yes Euler’s formula applies

5.

𝑣𝑣 = 6, 𝑒𝑒 = 9,𝑓𝑓 = 5

6 + 5 − 9 = 2, Euler’s formula applies to a triangular prism

6. Multiple answers, here are 2 examples

7. a) Multiple answers, here is one example:



b) A subgraph of the one in a) could be


1. a. 𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷,𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴,𝐴𝐴𝐵𝐵𝐷𝐷𝐴𝐴,𝐷𝐷𝐴𝐴𝐵𝐵𝐴𝐴 and others

b. 𝐴𝐴𝐷𝐷𝐴𝐴𝐴𝐴𝐵𝐵,𝐴𝐴𝐴𝐴𝐷𝐷𝐴𝐴𝐵𝐵 and others

c. 𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴,𝐴𝐴𝐷𝐷𝐴𝐴𝐴𝐴𝐴𝐴𝐷𝐷𝐵𝐵𝐴𝐴,𝐵𝐵𝐷𝐷𝐴𝐴𝐵𝐵, 𝑒𝑒𝑚𝑚𝑐𝑐.

d. 𝐴𝐴𝐴𝐴𝐷𝐷𝐵𝐵

e. 𝐴𝐴𝐷𝐷𝐵𝐵𝐴𝐴𝐴𝐴 is a closed path (of length 4)

f. 𝐴𝐴𝐷𝐷𝐴𝐴𝐴𝐴𝐵𝐵 is a trail because it travels along each edge once but it is not a path because it visits the vertex 𝐴𝐴 twice.

g. There are no bridges. Any edge may be removed from this graph and it would still stay be a connected graph.

2. a. There are many. 𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵,𝐵𝐵𝐸𝐸𝐴𝐴𝐷𝐷𝐸𝐸,𝐷𝐷𝐴𝐴𝐸𝐸𝐵𝐵𝐴𝐴 are some examples

b. There are many. 𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝐷𝐷,𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴 are some examples

c. 𝐴𝐴𝐵𝐵𝐴𝐴,𝐷𝐷𝐸𝐸𝐴𝐴

d. 𝐵𝐵𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴,𝐸𝐸𝐴𝐴𝐷𝐷𝐸𝐸𝐵𝐵𝐴𝐴 are two examples (no repeated edges)

e. 𝐵𝐵𝐸𝐸

f. 𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝐷𝐷 is an open path (no repeated edges or vertices)

g. 𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸𝐴𝐴𝐷𝐷𝐸𝐸 is a trail because it has no repeated edges. It is not a path because it has a repeated vertex, 𝐸𝐸.


1. 𝐴𝐴𝐷𝐷𝐴𝐴𝑋𝑋, 240 𝑚𝑚

2. 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑍𝑍, 55 𝑚𝑚

3. a) 𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐸𝐸, 3.7 𝑘𝑘𝑚𝑚 b) 𝐴𝐴𝐵𝐵𝐷𝐷𝐴𝐴𝐸𝐸, 3 𝑘𝑘𝑚𝑚




1.

2.

3. The graph is not traversable. All of its vertices are of degree 3 (odd). For a graph to be traversable, it must have 0 or 2 odd vertices.

4. 𝐸𝐸𝐴𝐴𝐴𝐴𝐵𝐵𝐴𝐴𝐷𝐷𝐵𝐵𝐸𝐸𝐷𝐷,𝐸𝐸𝐷𝐷𝐴𝐴𝐵𝐵𝐴𝐴𝐴𝐴𝐸𝐸𝐵𝐵𝐷𝐷 both start at 𝐸𝐸 and end at 𝐷𝐷.

𝐷𝐷𝐴𝐴𝐵𝐵𝐷𝐷𝐸𝐸𝐴𝐴𝐴𝐴𝐵𝐵𝐸𝐸,𝐷𝐷𝐵𝐵𝐴𝐴𝐴𝐴𝐸𝐸𝐷𝐷𝐴𝐴𝐵𝐵𝐸𝐸 both start at 𝐷𝐷 and end at 𝐸𝐸.

All paths start at either 𝐷𝐷 or 𝐸𝐸 and finish at the other. There may be more not listed here.

5. a. semi-Eulerian

b. Eulerian

c. Eulerian

6. a. 𝐵𝐵𝐴𝐴𝐷𝐷𝐴𝐴𝐵𝐵𝐸𝐸𝐷𝐷𝐴𝐴 (or its reverse), 𝐴𝐴𝐷𝐷𝐴𝐴𝐵𝐵𝐸𝐸𝐷𝐷𝐴𝐴𝐵𝐵. There may be more, all will start at 𝐵𝐵 and end at 𝐴𝐴 or vice-versa.

b) semi-Eulerian trail, there are odd vertices in the trail. It starts and ends at different vertices, i.e. is an open trail.

7.

Vertex Degree A 3 B 4 C 3 D 4 E 2

Vertex In-degree Out-degree P 0 3 Q 3 1 R 2 1



The following marking keys have been sourced from previous SCSA Examination Paper Ratified Marking Keys. Copyright © School Curriculum and Standards Authority 2019 https://senior-secondary.scsa.wa.edu.au/syllabus-and-support-materials/mathematics/mathematics-applications

2019 ATAR Calculator-Free Q1 Marking Key





2019 ATAR Calculator-Free Q4 (a) & (b) Marking Key




ATAR 2019 Calculator-Assumed Examination Question 12 (6 marks)




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Year 12 ATAR Mathematics Applications · 2020. 4. 21. · applications