The idea, step by step
Objects and connections
A network records relationships. A vertex represents an object, such as a town, station or computer. An edge represents a direct connection between two objects. The drawing need not preserve real distances: its essential information is which objects are joined.
AB and BA name the same edge in an undirected network. A route through another vertex is indirect and does not create a new direct edge. List edges once to avoid double-counting.
Degree and an edge-count check
The degree of a vertex is the number of edges meeting it. Count only edges with that vertex as an endpoint. A vertex with no incident edge has degree zero and is called isolated.
Every ordinary undirected edge contributes one to each endpoint, so the degree sum is twice the edge count. This gives a check on a crowded diagram.
A common trap
Counting the route A–B–C as a direct edge AC.
Instead: A route uses existing edges. AC is an edge only when A and C are joined directly.
See the thinking
Count a labelled network
Count vertices and edges and find the degree of B.
- A–E gives five vertices.
- The edges are AB, BC, AD, BD, BE, CE and DE.
- Four edges meet B: AB, BC, BD and BE.
Include an isolated vertex
The edges are PQ, QR, RP and a fourth vertex S is isolated. Find all degrees.
- P, Q and R each meet two edges.
- S meets none.
- The sum 2+2+2+0=6 equals twice the three edges.
Check your understanding
The degrees are 1,2,2,3,3,3. How many edges?
Show answer
7.
The sum is 14 and each edge contributes twice, so 14 ÷ 2 = 7.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Count the vertices and edges.2 marks
- List all neighbours of D.2 marks
- Find all degrees and check their sum.3 marks
- A network has degrees 2,2,2,2,0. Find its edge count.2 marks
- Can three vertices each have degree 1, with no other vertices? Explain.3 marks
- In a mutual-friendship network, what does degree 4 represent?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Count the vertices and edges.2 marks
Count A–E and list the seven distinct links.
5 vertices, 7 edges. - List all neighbours of D.2 marks
Read AD, BD, DE.
A, B, E. - Find all degrees and check their sum.3 marks
Count incident links.
14=2×7.
A2, B4, C2, D3, E3; sum 14. - A network has degrees 2,2,2,2,0. Find its edge count.2 marks
Sum=8.
8÷2=4.
4. - Can three vertices each have degree 1, with no other vertices? Explain.3 marks
The degree sum would be 3.
An undirected degree sum is even.
No. - In a mutual-friendship network, what does degree 4 represent?2 marks
Interpret one vertex as one person and its four edges as four direct relationships.
A person with four recorded mutual friendships.
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 ↗