How many SCSA Mathematics Applications questions cover Graphs and networks?

AusGrader has 126 SCSA Mathematics Applications questions on Graphs and networks, all with instant AI grading and detailed marking feedback.

WACE Mathematics Applications — Graphs and networks Questions & Answers

Q38

2024

VCAA

Paper 1

1 mark

Q38

1 mark

A connected graph has six vertices and six edges.

How many of the following four statements must always be true?

the graph has no vertices of odd degree
the graph contains a Eulerian trail
the graph contains a Hamiltonian path
the sum of the degrees of the vertices is 12

A

1

B

2

C

3

D

4

Reveal Answer

A

1

Correct Answer

Only one statement is always true: the sum of the degrees is 12. By the Handshaking Lemma, the sum of degrees is always twice the number of edges ( $2 \times 6 = 12$ ). The other statements can be disproven with counterexamples.

B

2

Only one statement is always true. A graph consisting of a triangle with three separate edges attached to its vertices has degrees of 3, 3, 3, 1, 1, and 1, which disproves the first three statements.

C

3

Only one statement is always true. A connected graph with 6 vertices and 6 edges does not necessarily have a Eulerian trail or a Hamiltonian path, nor does it require all even degrees.

D

4

It is impossible for all four statements to always be true. Only the statement regarding the sum of the degrees being 12 is a universal property of any graph with 6 edges.

Q8

2021

QCAA

Paper 1

1 mark

Q8

1 mark

A basketball competition has six teams that have completed three rounds of competition as shown.

	Bears	Eagles	Lions	Meerkats	Tigers	Wombats
Bears	—	—	✓	—	✓	✓
Eagles	—	—	✓	✓	—	✓
Lions	✓	✓	—	✓	—	—
Meerkats	—	✓	✓	—	✓	—
Tigers	✓	—	—	✓	—	✓
Wombats	✓	✓	—	—	✓	—

The graph to represent this information has

A

6 edges.

B

9 edges.

C

15 edges.

D

18 edges.

Reveal Answer

A

6 edges.

This number corresponds to the number of vertices (teams) in the graph, not the number of edges (games played).

B

9 edges.

Correct Answer

The table contains 18 checkmarks total. Since each game involves two teams and is recorded for both, the number of edges is half the total count: $18 \div 2 = 9$ .

C

15 edges.

This would be the number of edges if every team played every other team (a complete graph $K_6$ ), calculated as $\frac{6 \times 5}{2} = 15$ .

D

18 edges.

This is the total number of checkmarks in the table. Because the graph is undirected (Team A vs Team B is the same game as Team B vs Team A), you must divide this sum by 2.

Q32

2025

VCAA

Paper 1

1 mark

Q32

1 mark

Kyle ( $K$ ), Lian ( $L$ ), Maggie ( $M$ ), Neil ( $N$ ) and Ophelia ( $O$ ) took part in a round-robin chess tournament in which each person played each of the others once. In each game there was a winner and a loser.

The winner of the tournament was determined by $T = D + D^2$ where $D$ and $D^2$ are, respectively, the one-step and two-step dominance matrices.

Some of the individual match results were not recorded.

An incomplete matrix $D$ is shown below.

The '1' in row $K$ , column $M$ indicates that Kyle defeated Maggie.

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ D = \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & ... & 1 & ... & 0 \\ ... & 0 & 0 & ... & 0 \\ 0 & 1 & 0 & 1 & 1 \\ ... & ... & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \end{bmatrix} \end{matrix}$

The following information is known.

Maggie and Ophelia each won three of their four games.
Kyle won two of his four games.
Lian and Neil each won one of their four games.
Kyle defeated Neil.

Which one of the following is matrix $T$ ?

A

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 2 & 1 & 2 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 2 & 2 & 0 & 3 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 2 & 0 \end{bmatrix} \end{matrix}$

B

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 2 & 1 & 2 & 1 \\ 1 & 0 & 1 & 1 & 0 \\ 2 & 3 & 0 & 2 & 1 \\ 1 & 1 & 0 & 0 & 0 \\ 2 & 2 & 1 & 2 & 0 \end{bmatrix} \end{matrix}$

C

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \end{bmatrix} \end{matrix}$

D

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \end{bmatrix} \end{matrix}$

Reveal Answer

A

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 2 & 1 & 2 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 2 & 2 & 0 & 3 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 2 & 2 & 1 & 2 & 0 \end{bmatrix} \end{matrix}$

This matrix contains errors in calculating the two-step dominance matrix $D^2$ . For example, in row L, it misses Lian's two-step dominances over Maggie and Neil.

B

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 2 & 1 & 2 & 1 \\ 1 & 0 & 1 & 1 & 0 \\ 2 & 3 & 0 & 2 & 1 \\ 1 & 1 & 0 & 0 & 0 \\ 2 & 2 & 1 & 2 & 0 \end{bmatrix} \end{matrix}$

Correct Answer

By using the given win totals and match results, we can complete the one-step dominance matrix $D$ . Calculating $T = D + D^2$ yields this exact matrix, which correctly accounts for both direct wins and two-step dominances.

C

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \end{bmatrix} \end{matrix}$

This matrix does not represent $T = D + D^2$ . It appears to be an incorrectly filled one-step dominance matrix $D$ that does not align with the given win totals.

D

$\begin{matrix} & \begin{matrix} K & L & M & N & O \end{matrix} \\ \begin{matrix} K \\ L \\ M \\ N \\ O \end{matrix} & \begin{bmatrix} 0 & 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \end{bmatrix} \end{matrix}$

This is the completed one-step dominance matrix $D$ . However, the question asks for the total dominance matrix $T = D + D^2$ , which must also include the two-step dominances.

Q2

2024

QCAA

Paper 1

1 mark

Q2

1 mark

In a graph, an open walk with repeated vertices and no repeated edges is called a

A

bridge.

B

loop.

C

path.

D

trail.

Reveal Answer

A

bridge.

A bridge is a specific type of edge whose removal increases the number of connected components in a graph, not a type of walk.

B

loop.

A loop is an edge that connects a vertex to itself, rather than a sequence of vertices and edges known as a walk.

C

path.

A path is a walk where all vertices must be distinct. Since the question specifies that vertices are repeated, it cannot be a path.

D

trail.

Correct Answer

A trail is defined as a walk in which no edges are repeated, but vertices are allowed to be repeated.

Q34

2023

VCAA

Paper 1

1 mark

Q34

1 mark

A bipartite graph is typically used to display which one of the following?

A

the allocation of tasks on a construction site

B

the path used to visit five different construction sites

C

the total distance travelled between two construction sites

D

the critical path of activities to be completed in a construction project

E

the minimum length of cable required to connect six construction sites

Reveal Answer

A

the allocation of tasks on a construction site

Correct Answer

Bipartite graphs are ideal for modeling assignment problems, such as allocating a set of workers to a set of tasks, where edges connect the two distinct groups.

B

the path used to visit five different construction sites

Finding a path to visit multiple sites is a routing problem (like the Travelling Salesperson Problem), which is typically modeled using a complete or general graph.

C

the total distance travelled between two construction sites

Determining the distance between two sites is a shortest path problem, which is best modeled using a weighted graph rather than a bipartite graph.

D

the critical path of activities to be completed in a construction project

Critical path analysis for project activities is modeled using a directed acyclic graph (DAG) or an activity network, as it requires showing dependencies and sequence.

E

the minimum length of cable required to connect six construction sites

Finding the minimum cable to connect multiple sites is a minimum spanning tree problem, which is modeled using a weighted graph.

SCSA Mathematics Applications Graphs and networks

Frequently Asked Questions

Ready to practise SCSA Mathematics Applications?