What is a weighted graph?

A weighted graph is a graph in which every edge is assigned a weight (a numeric value).

Edge weight is a characteristic of the connection between vertices: distance, cost, time, connection strength, and so on.

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#Formally

A weighted graph is G = (V, E, w), where w(u, v) is a function that defines the weight of edge (u, v).

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#Example

Let there be vertices - cities: A, B, C, and edges with distances:

• A-B (5 km)
• B-C (3 km)
• A-C (8 km)

Then the edge weights are 5, 3, and 8.

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#Applications

• Maps and navigation (weight = distance or time).
• Cost optimization (weight = price).
• Shortest-path algorithms (Dijkstra, Bellman-Ford).

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Summary: a weighted graph is a graph where every edge has a weight that reflects the "cost" of the connection between vertices.