The idea, step by step
Drawing shape is not network structure
You can move vertices and bend edges without changing a network, provided the endpoints remain the same. Compare labelled diagrams by their vertex sets and edge sets, not by whether one looks square and another looks curved.
Degree lists provide a quick check but are not a complete comparison. Two labelled networks can give every vertex the same degree while connecting different pairs. A claim that networks are identical needs the actual connection matching.
Crossings and repeated entries
Lines crossing on a page do not automatically create a junction. There must be a marked vertex or a clear statement that travel can switch links at that point. A road bridge over another road is a practical example.
In a simple undirected connection table, row sums give degrees. Summing the whole table counts every edge twice. A table can reveal details hidden by a crowded drawing.
A common trap
Counting every crossing as a vertex.
Instead: Use marked vertices and the stated junction rule.
See the thinking
Reverse edge names
P has AB, BC, CD, DA. Q has DA, CB, BA, DC. Are the labelled networks identical?
- In an undirected graph CB=BC, BA=AB and DC=CD.
- The edge sets match exactly.
A crossing without a junction
Can a traveller go from A to B in the crossing-edge diagram?
- A is joined only to D.
- B is joined only to C.
- No vertex permits switching at the crossing.
Check your understanding
A symmetric zero-diagonal table contains twelve 1s. How many edges?
Show answer
6.
Each undirected edge creates two symmetric entries.
Your turn
6 questions · 14 marks · Work at your own pace.
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 ↗