Understand the idea.
Drawing shape is not network structure
A network’s structure depends on its connections, not its outline. Moving vertices, bending edges or changing the layout preserves a labelled network when every edge keeps the same endpoints.
Compare vertex sets and edge sets to decide whether two labelled representations agree. Matching degrees is a useful check but does not prove agreement: different pairs can be joined while leaving every vertex’s degree unchanged.
Crossings and repeated entries
A crossing is a junction only if a vertex is marked there or the situation states that switching is possible. Two roads crossing at different heights illustrate why intersecting lines need not create a connection.
A connection table provides another way to check a diagram. For a simple undirected network, it is symmetric, each row sum gives a degree, and the total of all entries counts every edge twice.
See the method in action.
Reverse edge names
P has AB, BC, CD, DA. Q has DA, CB, BA, DC. Are the labelled networks identical?
Worked solution
- 1
Use the undirected-edge convention.
Edge names may be reversed without changing their endpoints.
- 2
Write both edge sets in the same order.
The reversed names and different listing order describe exactly the same four links.
- 3
Conclude that the labelled networks agree.
Both vertex sets and both edge sets match, so the networks are identical.
Yes.
A crossing without a junction
Can a traveller go from A to B in the crossing-edge diagram?
Worked solution
- 1
Trace the component containing A.
AD is the only edge incident to either of these vertices.
- 2
Trace the component containing B.
BC is a separate edge; the crossing has no marked vertex.
- 3
Check whether switching is allowed.
There is no junction at the crossing, so no route can leave the A–D component to reach B.
No.
Try it yourself.
Check your understanding
A symmetric zero-diagonal table contains twelve 1s. How many edges?
Show answer & reasoning
6.
Each undirected edge produces two symmetric entries, and the zero diagonal excludes self-links. Therefore twelve entries represent edges.
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- A diagram is redrawn with all vertices moved but the same labelled edge set. Has the network changed?2 marks
- Count vertices and components in the crossing-edge diagram.2 marks
- An undirected table has row sums 2,3,1,2. Find the number of edges.2 marks
- P has AB, BC, CD, DA. Q has AC, CB, BD, DA. Are these the same labelled network?3 marks
- Stations move farther apart on a non-scale diagram. Can you infer longer travel time?2 marks
- List edges from the table and say whether it is connected.3 marks
From / to A B C D A 0 1 0 1 B 1 0 1 0 C 0 1 0 1 D 1 0 1 0
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A diagram is redrawn with all vertices moved but the same labelled edge set. Has the network changed?2 marks
The direct relationships are unchanged.
No. - Count vertices and components in the crossing-edge diagram.2 marks
The crossing is unmarked.
The components are {A,D} and {B,C}.
4 vertices and 2 components. - An undirected table has row sums 2,3,1,2. Find the number of edges.2 marks
The sum is 8; half is 4.
4. - P has AB, BC, CD, DA. Q has AC, CB, BD, DA. Are these the same labelled network?3 marks
AB occurs only in P.
All degrees are 2 in each network, so matching degrees alone is insufficient.
No. - Stations move farther apart on a non-scale diagram. Can you infer longer travel time?2 marks
Page distances do not represent time unless that rule is stated.
No. - List edges from the table and say whether it is connected.3 marks
From / to A B C D A 0 1 0 1 B 1 0 1 0 C 0 1 0 1 D 1 0 1 0 The edges form A–B–C–D–A.
AB, AD, BC, CD; connected.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum
Vertices, edges and connections; drawing networks from situations; interpreting connectedness and routes; comparing representations; using networks to explain practical relationships.
Official curriculum source ↗