1. © Nuffield Foundation 2012 Nuffield Free-Standing Mathematics Activity Networks © Rudolf Stricker

2. Networks This diagram shows a network. Problems involving the study of networks include optimisation tasks such as finding the shortest route and minimum cost. This activity introduces some of the terms and methods used.

3. vertex (node) edge Vertices can be even or odd or undirected (two-way) Edges can be directed (one-way) This is an example of a graph. This graph is connected Think about … Why must the sum of the degrees of the vertices in any graph always be even? What can you say about the number of odd vertices in a graph? The degree of a vertex is the number of edges that meet there

4. 49 mins 39 mins 28 mins 22 mins 24 mins 29 mins 34 mins 15 miles 17 miles 15 miles 12 miles 17 miles 21 miles 20 miles Networks Colchester Stowmarket Ipswich Sudbury Bury St Edmunds Harwich A network is a weighted graph. The weights could be distances, times or costs.

5. Paths Paths are routes that do not visit any vertex more than once and do not go along any edge more than once. A cycle forms a loop by returning to its starting point.

6. Colchester Stowmarket Ipswich Sudbury Bury St Edmunds Harwich Adjacency matrix B Su C H I St B Su C H I St

7. Colchester Stowmarket Ipswich Sudbury Bury St Edmunds Harwich Distance matrix B Su C H I St B Su C H I St 15 miles 17 miles 15 miles 12 miles 17 miles 21 miles 20 miles Think about… What do you notice about the pattern in these matrices?

8. Reflect on your work Explain what is meant by the following terms: network, graph, edge, vertex, node, degree, directed, undirected, weighted, path, cycle, connected, adjacency matrix, distance matrix. © Nuffield Foundation 2012 If a road is one way, then when planning a driving route it may be possible to get from A to B, but not from B to A. What difference would this make to the adjacency matrix?