Year 12 Graph Theory: Vertex properties

Vertex questions are the entry point to most graph-theory problems. Before you can analyse a network as a whole, you need to understand what is happening locally at each vertex. Degree, adjacency, isolation, and incidence all describe how a vertex sits inside the graph. These ideas are simple, but they matter because many later properties such as paths, circuits, and even network feasibility depend on them. Strong Year 12 graph theory begins by reading vertex information accurately and translating the picture into clean vocabulary.

Treat each vertex as a local summary of the network. Degree and adjacency tell you how that point is connected before you start reasoning globally.

Subtopic 1: Degree of a vertex

The degree of a vertex is the number of edges incident with it. If a vertex has many connections, its degree is high. If it has no connections, it is isolated and has degree zero. Degree is often the first numerical feature you should look for because it immediately tells you something about how much movement, contact, or branching is possible at that point in the network.

Worked example 1

Problem: A vertex

V

is joined to four other vertices by four distinct edges. Find the degree of

V

The degree counts how many edges meet at the vertex.
There are four edges incident with $V$ .

Answer: the degree of

V

4

Subtopic 2: Adjacency and neighbourhoods

Two vertices are adjacent if an edge directly joins them. This idea sounds basic, but it is essential because many path questions, matrix models, and graph descriptions are really just organised ways of recording adjacency. A vertex's neighbourhood is the collection of vertices adjacent to it. When you list the neighbours of a vertex, you are describing its immediate local environment in the graph.

Worked example 2

Problem: In a graph, vertex

A

is connected to vertices

B

D

, and

E

. State the vertices adjacent to

A

and the degree of

A

The adjacent vertices are exactly those joined directly to $A$ .
So the neighbours are $B$ , $D$ , and $E$ .
Since three edges meet at $A$ , the degree is $3$ .

Answer: the vertices adjacent to

A

are

B

D

, and

E

, and

deg (A) = 3

Why vertex information matters

Vertex properties often look too small to be important, but they are the building blocks for bigger results. For example, if every vertex has even degree, that immediately suggests certain kinds of circuit behaviour may be possible. If one vertex has extremely high degree, it may act as a hub in a transport or communication network. If several vertices are isolated, the graph cannot be fully connected. Local structure often predicts global structure.

Worked example 3

Problem: Explain what it means if a graph has an isolated vertex.

An isolated vertex has degree zero.
That means no edges are incident with it.
So it is not directly connected to any other vertex in the graph.

Answer: an isolated vertex is a vertex with no direct connections at all.

Common traps

Confusing the degree of a vertex with the total number of edges in the graph.
Forgetting that adjacency means direct connection, not indirect reachability.
Miscounting the degree when the diagram is crowded or badly drawn.
Describing neighbours informally without naming the relevant vertices clearly.

Revision focus

A strong revision habit is to relabel messy diagrams neatly and list each vertex with its degree before trying to solve anything larger. This small table often exposes patterns such as leaves, hubs, isolated points, or symmetry. Once the vertex data is explicit, many later arguments become much shorter.

It also helps to practise speaking graph language out loud. Saying "vertex $A$ is adjacent to $B$ , $D$ , and $E$ " is much more precise than saying "it connects to a few others." Graph theory rewards exact description, and this topic is where that habit starts.

Practice links

Properties of vertices