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A network of 52 sports training facilities in Melbourne is connected by direct delivery routes for equipment. Each facility is linked to exactly 28 others via these routes.
Using the principle that the sum of all vertex degrees equals twice the number of edges, determine the exact number of delivery routes in the network.
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Worked solution
- The network consists of 52 sports training facilities in Melbourne, each represented as a vertex in a graph.
- Each facility is connected to exactly 28 others via direct delivery routes, meaning each vertex has a degree of 28.
- The sum of all vertex degrees in a graph is equal to twice the number of edges (delivery routes). This is a fundamental principle in graph theory.
- Calculate the total sum of vertex degrees: 52 vertices × 28 degrees per vertex = 1456.
- Apply the formula: sum of degrees = 2 × number of edges. Therefore, 1456 = 2 × E.
- Solve for E by dividing both sides by 2: E = 1456 ÷ 2 = 728.
- The result is 728 edges, which represent the total number of direct delivery routes in the network.
- Check: 728 edges × 2 = 1456 total degrees, which matches the sum of 52 × 28. The calculation is consistent.
Reference answer: 728
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