A simple graph has no loops and no multiple edges; a complete graph connects every pair of distinct vertices; and a weighted graph assigns a numerical value to each edge. These categories can overlap, so they are not three mutually exclusive graph types.
This is AHL 3.14: Graph theory, so it is HL-only content. A graph consists of vertices joined by edges.
| Graph type | Defining feature | Key exam check |
|---|---|---|
| Simple graph | No edge joins a vertex to itself, and no pair of vertices has more than one connecting edge | Check for loops and repeated edges |
| Complete graph | Every pair of distinct vertices is connected by exactly one edge | Each vertex must have degree |
| Weighted graph | Every edge has an associated number, such as distance, cost, or time | Include and interpret the edge weights |
A complete graph with vertices is denoted by . Each vertex has degree , and the total number of edges is
For example, has
edges, and every vertex has degree . An ordinary is both complete and simple. If numerical distances are added to its six edges, it is also weighted.
The common misconception is that “simple,” “complete,” and “weighted” must describe separate categories. In fact, simple describes restrictions on edges, complete describes how fully the vertices are connected, and weighted describes whether edges carry values.
Exam technique: In an IB Math AI HL Paper 1 short-response question, you may be asked to identify or determine the graph type. Justify your answer using the defining property rather than appearance alone: mention loops or repeated edges, connection of every vertex pair, or numerical edge weights. Do not assume that a weighted graph cannot also be simple or complete.