#MeasurementAndSpace #Path #Stn >[!info]- [Introduction to networks (Path) | NSW curriculum Website](https://curriculum.nsw.edu.au/learning-areas/mathematics/mathematics-k-10-2022/content/stage-5/fae172f5e1) >- MA5-NET-P-01 solves problems involving the characteristics of graphs/networks, planar graphs and Eulerian trails and circuits ## 📖 Prior Knowledge: None! ## Examine and describe a graph/network - Describe a network as a collection of objects (nodes or vertices) interconnected by lines (edges) that can represent systems in the real world - Examine real-world applications of networks such as social networks, supply chain networks and communication infrastructure, and explore other applications of networks - Explain that the terms _graph_ and _network_ are interchangeable - Identify and define elements of a graph including vertex, edge and degree - Explain that a given graph can be drawn in different ways ## Define a planar graph and apply Euler’s formula for planar graphs - Define a planar graph as any graph that can be drawn in the plane so that no 2 edges cross - Define a non-planar graph as a graph that can never be drawn in the plane without some edges crossing - Demonstrate that some graphs that have crossing edges are still planar if they can be redrawn so that no 2 edges cross - Identify the number of faces in a planar graph - Describe and apply Euler’s formula for planar graphs: $v-e+f=2$, where $v=$ vertices, $e=$ edges and $f=$ faces ## Explain the concept of Eulerian trails and circuits in the context of the Königsberg bridges problem - Explain that a connected graph is a graph that is in one piece, so that any 2 vertices are connected by a path - Define a walk on a graph to be a sequence of vertices and edges of a graph - Explain the difference between trail, circuit, path and cycle - Relate the definition of a trail to an Eulerian trail as a walk in which every edge in the graph is included exactly once - Relate the definition of a circuit to an Eulerian circuit that is defined as an Eulerian trail that ends at its starting point - Relate Euler’s Seven Bridges of Königsberg network problem to the definition of an Eulerian trail or circuit