A network is made of vertices joined by edges. The degree of a vertex is the number of edges that meet at it.
This lesson opens the SPM Mathematics networks chapter. The next lesson shows how to draw real situations as networks.
What do the words mean?
| Word | Meaning | Drawn as |
|---|---|---|
| Vertex | A point in the network | A labelled dot |
| Edge | A connection between two vertices | A line |
| Degree | Number of edges meeting at a vertex | A number you count |
Each edge has exactly two ends, and each end sits on a vertex.
Worked example: counting degrees
A network has vertices A, B, C, D and E. The edges are AB, AC, BC, CD and DE.
Count how many edges touch each vertex:
| Vertex | Edges touching it | Degree |
|---|---|---|
| A | AB, AC | 2 |
| B | AB, BC | 2 |
| C | AC, BC, CD | 3 |
| D | CD, DE | 2 |
| E | DE | 1 |
The degrees add to 2 + 2 + 3 + 2 + 1 = 10. There are 5 edges, and 2 × 5 = 10, so the count passes the check.
The mistake that costs marks
The common slip is to count each edge only once across the whole table. A student ticks each edge off once, so an edge counted for B is never counted again for C. The fix is to count from each vertex’s point of view.
| Wrong | Right | |
|---|---|---|
| Method | Tick off each edge once | For each vertex, list all edges touching it |
| Degrees | 2, 1, 3, 1, 1 (total 8) | 2, 2, 3, 2, 1 (total 10) |
| Check | Total 8 is not 2 × 5 | Total 10 equals 2 × 5 |
When the total is not twice the edge count, recount. The error is always in the degree list.
Check yourself
A network has vertices P, Q, R and S, with edges PQ, PR, PS and QR. Find the degree of each vertex and check your counts.
Answer
P is on PQ, PR and PS, so its degree is 3.
Q is on PQ and QR, so its degree is 2. R is on PR and QR, so its degree is 2. S is on PS only, so its degree is 1.
Sum = 3 + 2 + 2 + 1 = 8. There are 4 edges and 2 × 4 = 8, so the counts are correct.
What to study next
Move on to representing a real network as a graph. If your slips come from careless working in general, the algebra step-repair trainer builds the habit of checking each step. Log slips in the mistake log and paper-error review.
If you want a teacher to watch you count, see online one-to-one Mathematics tuition.