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

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Required skills (click to reveal)

A school's order network for classroom supplies connects 60 departments (nodes) via direct communication lines (edges). Each department is connected to exactly 85 other departments.

Using the principle that the sum of all vertex degrees equals twice the number of edges, determine the exact number of communication lines 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 60 departments, each represented as a node in a graph.
  2. Each department is connected to exactly 85 other departments, meaning each node has a degree of 85.
  3. The sum of all vertex degrees in a graph is equal to twice the number of edges, according to the Handshaking Lemma.
  4. Calculate the total sum of degrees: 60 nodes × 85 degrees per node = 5100.
  5. Apply the formula: sum of degrees = 2 × number of edges.
  6. Substitute the sum of degrees: 5100 = 2 × E.
  7. Solve for E by dividing both sides by 2: E = 5100 ÷ 2.
  8. Perform the division: 5100 ÷ 2 = 2550.
  9. The result is 2550 edges, which represent the communication lines.
  10. Check: 2 × 2550 = 5100, which matches the sum of degrees.
  11. Therefore, the network has exactly 2550 communication lines.
  12. The solution is consistent with graph theory principles and the given data.

Reference answer: 2550

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