Year 12: Graph theory

Graph theory questions are about extracting structure from a network. Once you identify vertices, edges, degrees and paths clearly, the question usually becomes a sequence of short observations.

Translate the picture into graph vocabulary first. Naming the properties you see is usually the fastest route to the correct method.

What to master

Describing vertex properties such as degree and adjacency.
Recognising overall graph properties and what they imply.
Finding shortest paths efficiently and checking all relevant candidates.
Explaining solutions using correct graph-theory language.

Exam habit: if several routes have similar lengths, tabulate them instead of relying on visual inspection. The shortest path is easy to misread from a crowded diagram.

Graph Theory tutorials

How to work through a network

Graph theory rewards precise vocabulary. Once you identify the vertices, edges, degrees, adjacency, and path lengths clearly, many questions reduce to a sequence of small decisions. That is why graph theory often feels easier after you translate the picture into language first. The drawing matters, but the real mathematical work comes from describing what the drawing is telling you in exact terms.

Worked example

Problem: A vertex is joined to four other vertices in a graph. What is the degree of that vertex, and what does the answer mean?

The degree of a vertex is the number of edges attached to it.
If the vertex is joined to four others, then four edges meet at that vertex.
So the degree is 4.

Answer: the degree is

4

, meaning four edges are incident with that vertex.

Common traps

Confusing the degree of a vertex with the total number of edges in the graph.
Assuming the visually shortest route is the true shortest weighted path.
Describing a graph property informally without using graph-theory language.

Study routine

A strong revision habit is to redraw a graph more cleanly before solving the problem. Reorganising the layout does not change adjacency, but it often makes routes and degree patterns easier to see. That is especially helpful on crowded shortest-path questions.

It is also worth checking whether the question is about the shape of the network or about weights on the network. A path can use fewer edges but still have a larger total weight. Keeping that distinction clear helps you decide whether you should count connections, compare distances, or analyse a property of the graph as a whole.

Revision focus

These topics reward careful structure more than fast calculation. In complex numbers, discrete mathematics, graph theory, and number theory, the key objects carry meaning that should stay visible all the way through the working. Real and imaginary parts, matrix entries, graph properties, and modular statements all behave best when they are written clearly and interpreted consistently.

A reliable way to revise is to explain each line of working in words as you go. If you can say what a power of i is doing, what a matrix row represents, what a graph property implies, or what a congruence statement means, then the notation is working for you rather than against you. That level of clarity usually prevents the small symbolic errors that turn easy marks into lost marks.