Network in Graph Theory · Form 4
What does a weighted network model?
A weighted network adds a number to each edge, a distance, time or cost, so the diagram measures the connection, not just shows that one exists.
From 'connected' to 'how much'
A plain network only tells you what is joined to what. A weighted network puts a number on each edge, a distance in km, a travel time in minutes, or a cost in ringgit, so the picture now measures the connection, not just records it.
The number attached to an edge is called its weight.
What the weights let you do
Once edges carry weights, you can add them along a route and compare one route with another, for example, which path between two towns is shortest, or which set of cables is cheapest. You recognise a weighted network instantly: numbers sit on the edges, not just labels on the vertices.
Real questions almost always dress this up as maps, pipelines or delivery routes.
Weight is not degree
Keep two ideas apart: weight belongs to an edge and says how far or how costly, while degree belongs to a vertex and counts lines. A short edge can still touch a high-degree vertex.
Note too that weighting and direction are separate features, an edge can have a weight, an arrow, both, or neither, so read which the question is using.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How are weighted networks used in shortest-route exam questions?
You usually find the shortest, cheapest, or fastest route between two vertices by comparing the total weight of different possible paths. List each possible route, add up the weights of its edges, then compare the totals to pick the best one.
What's a common mistake when adding up weights along a route?
Adding the weight of an edge that isn't actually on your chosen path, or missing an edge partway through a route with several vertices. Trace the path vertex by vertex on the diagram, ticking off each edge, so none is skipped or double-counted.
What real situations does a weighted network's number usually represent?
Weights usually represent something measurable along the connection, such as the distance in kilometres between towns, the travel time along a road, or the cost of a delivery route. Always check the diagram's key or the question's wording to see what the weight stands for.