Network in Graph Theory · 5.1.2
Directed and weighted graphs
Students distinguish graphs whose edges show a specific direction, using arrows, from those whose edges show a two-way connection with no direction. They also distinguish graphs whose edges carry a numerical value, such as distance or cost, from those whose edges show only a connection with no value.
The official learning standard (5.1.2)
“Compare and contrast: (i) directed graphs and undirected graphs, (ii) weighted graphs and unweighted graphs.”
What it means
Students distinguish graphs whose edges show a specific direction, using arrows, from those whose edges show a two-way connection with no direction. They also distinguish graphs whose edges carry a numerical value, such as distance or cost, from those whose edges show only a connection with no value.
How it is examined
Paper 1 tests this mainly through multiple-choice items asking students to classify a given graph diagram correctly. In Paper 2, students usually classify the type of graph as an early sub-question, before using its properties, such as edge weights, to solve the rest of a network problem.
Worked example
Diagram shows a graph with vertices K, L and M. Bus route K to L, L to M, and M to K each run in one direction only, shown by arrows, and are labelled with their distances: KL = 5 km, LM = 3 km, MK = 4 km.
State, with reasons, whether the graph is (a) directed or undirected, (b) weighted or unweighted.
- (a) Each route is shown with an arrow indicating travel in only one direction (K→L, L→M, M→K, with no reverse routes given), so the graph is a directed graph.
- (b) Each edge is labelled with a numerical distance value (5 km, 3 km, 4 km), so the graph is also a weighted graph.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Can a graph be undirected but still weighted?
Yes. Direction and weight are independent properties.
An undirected graph, where edges show a two-way connection with no arrows, can still have numbers labelling each edge, for example, a map showing distances between towns with two-way roads is undirected but weighted.
How can I tell if a real-life network should be modelled as directed?
Ask whether movement or flow only works one way. A one-way street, a water pipe flowing in a single direction, or a task that must happen before another suggests a directed graph.
If travel or connection works equally in both directions, it should be modelled as undirected.
What kind of value can appear as a weight on an edge?
A weight is usually a real quantity relevant to the situation, such as distance in kilometres, travel time, or cost in ringgit. The specific unit depends on the context of the network described in the question, but it must always be a meaningful number, not just a label.