Understand the idea.
Objects and connections
A vertex represents an object; an edge represents a direct connection. In a town network, the drawing records which towns have a link, while the spacing on the page need not represent actual distance.
In an undirected network, AB and BA denote one edge. A route from A through B to C uses two edges; it does not imply a direct edge AC. List each pair once when counting connections.
Degree and an edge-count check
The degree of a vertex counts the edges ending there. Trace each edge to its endpoints rather than counting nearby lines. An isolated vertex still belongs to the network and has degree .
Each undirected edge contributes twice to the total degree, once at each endpoint. Therefore divide the degree sum by to recover the edge count; an odd total signals a counting or recording error.
See the method in action.
Count a labelled network
Count vertices and edges and find the degree of B.
Worked solution
- 1
Count the labelled vertices.
Each labelled point is one vertex.
- 2
List each undirected edge once.
Reversing an edge’s name does not create another connection.
- 3
Count edges incident to B.
AB, BC, BD and BE each have B as an endpoint.
5 vertices, 7 edges; degree(B)=4.
Include an isolated vertex
The edges are PQ, QR, RP and a fourth vertex S is isolated. Find all degrees.
Worked solution
- 1
Count the edges meeting P, Q and R.
The three edges PQ, QR and RP form a triangle, with two incident edges at each vertex.
- 2
Include the isolated vertex.
S is present but has no incident edge.
- 3
Check the degree sum.
The sum counts both ends of each of the three edges.
P=2, Q=2, R=2, S=0.
Try it yourself.
Check your understanding
The degrees are 1,2,2,3,3,3. How many edges?
Show answer & reasoning
7.
The degrees sum to . Each undirected edge contributes two, so the edge count is .
Keep practising
6 worksheet questions · 14 marks
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.
.
A2, B4, C2, D3, E3; sum 14. - A network has degrees 2,2,2,2,0. Find its edge count.2 marks
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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 ↗