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

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In a community garden in Adelaide, a network diagram represents the connections between 71 planting beds (nodes) and 49 water access points (nodes) via irrigation channels (edges). Each planting bed is connected to exactly 2 water access points, and each water access point is connected to exactly 7 planting beds.

Using the principle that the sum of all degrees in a network equals twice the number of edges, determine the exact number of irrigation channels (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 71 planting beds (nodes) and 49 water access points (nodes), with irrigation channels (edges) connecting them.
  2. Each planting bed is connected to exactly 2 water access points, so the degree of each planting bed node is 2.
  3. The total degree contribution from all planting beds is 71 × 2 = 142.
  4. Each water access point is connected to exactly 7 planting beds, so the degree of each water access point node is 7.
  5. The total degree contribution from all water access points is 49 × 7 = 343.
  6. The sum of all degrees in the network is 142 + 343 = 485.
  7. According to the principle that the sum of all degrees in a network equals twice the number of edges, we have: sum of degrees = 2 × number of edges.
  8. Substitute the sum of degrees: 485 = 2 × E, where E is the number of edges (irrigation channels).
  9. Solve for E: E = 485 ÷ 2 = 242.5.
  10. Since the number of edges must be a whole number, and 242.5 is not an integer, this indicates a contradiction in the given conditions.
  11. Recheck the degree sum: 71 × 2 = 142, 49 × 7 = 343, total 485. 485 is odd, but twice the number of edges must be even. Therefore, no such network can exist with these parameters.
  12. The problem states to determine the number of irrigation channels using the degree sum principle. However, since the sum of degrees is odd, it violates the handshaking lemma, meaning no such network is possible. Therefore, the correct answer is that the number of edges cannot be determined under these constraints.

Reference answer: The network cannot exist with the given conditions because the sum of degrees is odd, violating the handshaking lemma.

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