NavGanitMateby NavGreen Compute, AustraliaBased on Australian Curriculum: Mathematics v9.0
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Networks and connectedness

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A sports equipment distribution network in Perth connects 81 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 17 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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Degree counts incident edges. This connected path has edges AB and BC. The degrees are 1 at A, 2 at B and 1 at C. Every pair of vertices is connected by a path.ABCdegree 1degree 2degree 1
This connected path has edges AB and BC. The degrees are 1 at A, 2 at B and 1 at C. Every pair of vertices is connected by a path.

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Worked solution

  1. The network has 81 training centres, each represented as a node. Each node has a degree of 17, meaning each centre is directly connected to 17 other centres.
  2. 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.
  3. Calculate the total sum of degrees: 81 nodes × 17 connections per node = 1377.
  4. Set up the equation: sum of degrees = 2 × E. Substitute the sum: 1377 = 2 × E.
  5. Solve for E by dividing both sides by 2: E = 1377 ÷ 2.
  6. Perform the division: 1377 ÷ 2 = 688.5.
  7. Since the number of edges must be a whole number (each route connects exactly two centres), a non-integer result indicates a contradiction.
  8. Re-evaluate the assumption: a network where every node has degree 17 and there are 81 nodes must have an even sum of degrees. However, 81 × 17 = 1377 is odd.
  9. The sum of degrees in any undirected graph must be even because each edge contributes 2 to the total degree count (one to each end). An odd sum is impossible.
  10. Therefore, it is mathematically impossible for a network with 81 nodes to have every node connected to exactly 17 others.
  11. The premise of the network cannot exist under standard graph theory rules. Thus, no valid number of edges satisfies the given conditions.
  12. The problem as stated contains a contradiction. No such network with 81 nodes and each node degree 17 can exist.

Reference answer: No such network can exist

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