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A sports equipment distribution network in Brisbane connects 51 training centres (nodes) through direct delivery routes (edges). Each route connects exactly two centres, and the network is designed so that every centre is directly connected to exactly 15 other centres.
Using the principle that the sum of all degrees in a network equals twice the number of edges, determine the exact number of delivery routes (edges) in the network.
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Worked solution
- The network has 51 training centres, each represented as a node. Each node has a degree of 15, meaning each centre is directly connected to 15 other centres.
- The sum of all degrees in a network is equal to twice the number of edges (E), according to the Handshaking Lemma in graph theory.
- Total degree sum = number of nodes × degree per node = 51 × 15.
- Calculate 51 × 15: 50 × 15 = 750, and 1 × 15 = 15, so 750 + 15 = 765.
- Total degree sum = 765. This equals 2 × E, where E is the number of edges (delivery routes).
- Solve for E: E = 765 ÷ 2.
- 765 ÷ 2 = 382.5. However, the number of edges must be an integer because edges are physical connections.
- A non-integer result contradicts the assumption that edges are whole connections. Therefore, a network with 51 nodes each of degree 15 cannot exist.
- The degree sum must be even for a valid undirected graph. 765 is odd, so it violates the Handshaking Lemma.
- Since the sum of degrees is odd, no such network can exist with the given parameters.
- The problem states the network is designed so that every centre connects to exactly 15 others, but this leads to a contradiction because 51 × 15 = 765 is odd.
- Therefore, no valid number of edges exists under these conditions. The network as described cannot exist.
Reference answer: The network cannot exist because the sum of degrees is odd.
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