Networks — NSW Year 10 Advanced Maths
Degrees, walks, trails and paths, Eulerian trails and circuits, and planar graphs with Euler's formula. Three subtopics and 60 practice questions, each with a fully worked solution, aligned to NESA outcome MA5-NET-P-01.
Networks in NSW Year 10 Advanced
Networks model connections, such as roads between towns, flights between cities or friends on social media, as vertices joined by edges. Year 10 Advanced students describe networks using degrees and types of routes, decide when every edge can be travelled exactly once, and use Euler's formula \(v-e+f=2\) for graphs that can be drawn with no edges crossing.
The three subtopics below cover Eulerian trails and circuits, the language and properties of networks, and planar graphs and Euler's formula. Networks is one of the 16 topics in NSW Year 10 Advanced Maths and sits within the Measurement and Space strand (NESA outcome MA5‑NET‑P‑01). It is Path content that leads to the networks and critical path topics in Mathematics Standard in Years 11 and 12.
Subtopics
Networks: common questions
What is an Eulerian circuit?
An Eulerian circuit is a route that uses every edge of a network exactly once and returns to its starting vertex. A connected network has one exactly when every vertex has even degree.
When does a network have an Eulerian trail?
A connected network has an Eulerian trail that is not a circuit when exactly two vertices have odd degree. The trail must start at one of these odd vertices and finish at the other.
What is Euler's formula for planar graphs?
For a connected planar graph, \(v-e+f=2\), where \(v\) is the number of vertices, \(e\) the number of edges and \(f\) the number of faces, including the outside face. A graph with 8 vertices and 14 edges has \(f=2-8+14=8\) faces.
What is the difference between a trail and a path?
A trail never repeats an edge but may revisit a vertex. A path never repeats a vertex, so it cannot repeat an edge either. A closed trail is a circuit and a closed path is a cycle.
Which NESA outcome covers Networks?
Networks are outcome MA5‑NET‑P‑01, a Path outcome in Stage 5 of the NSW Mathematics K–10 Syllabus (2022) that prepares students for networks in Mathematics Standard.