Year 10 Maths • AC9M10SP02 • Authored homework
Networks, vertices and connectedness
Interpret networks by defining vertices, edges, paths, degrees, weights and connectedness, then connect the diagram back to the practical situation it represents.
Part A
Short-answer questions
In a network diagram, what is a vertex?
In a network diagram, what is an edge?
What does it mean for a network to be connected?
What is the degree of a vertex?
A vertex has edges to four other vertices. What is its degree?
Explain the difference between a connected network and a complete network.
In a transport network, what could vertices and edges represent?
What does an edge weight usually represent?
State Euler’s formula for a convex polyhedron using V, E and F.
Give one example of a directed relationship that could be shown in a network.
Part B
Long-answer questions
A town bus network has stops A, B, C, D and E. Direct routes are AB, AC, BD, CD and DE. List the degree of each vertex and decide whether the network is connected.
A delivery driver can travel A–B = 5 km, A–C = 8 km, B–D = 4 km, C–D = 3 km and D–E = 6 km. Compare two routes from A to E and identify the shorter weighted route.
Draw or describe a network for a school friendship or communication situation. Define what the vertices and edges mean, then explain one limitation of the model.
A network has two connected components: {A, B, C} and {D, E}. Explain why the full network is disconnected and suggest one new edge that would make it connected.
Verify Euler’s formula for a cube using V = 8, E = 12 and F = 6. Then explain why the formula is about polyhedra, not every network diagram.
A home internet network has a modem connected to two routers, and each router connects to several devices. Represent this as a network and explain how one router failure could affect connectedness.
Compare a physical map of train lines with a network diagram of the same system. Explain what information the network keeps and what information it may ignore.
A food web is represented as a directed network. Explain what the vertices, directed edges and connected paths could mean, and why direction matters.
A network has vertices A, B, C, D and edges AB, BC, CD and DA. Add one edge that changes the degree of two vertices but keeps the network connected. Explain the change.
Create a practical network problem involving transport, communication, electricity or sport fixtures. Include vertices, edges, any weights, a question about connectedness or route choice, and a full solution.
Part C
Research and understanding task
Choose a real network such as a bus route, school device network, sports draw, food web or family/contact network. Define the vertices and edges, sketch or describe the network, identify degrees and connectedness, and explain what the model reveals and what it leaves out.