Undirected graphs and networks

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154 General Mathematics Undirected graphs and networks Graphs are an efficient way of summarising data in many practical problems. The graphs we will be dealing with in this chapter differ from those that we have worked with in the past, as they consist of points connected by various lines. As there is no particular order or direction to these lines, the graphs are defined as undirected graphs or networks. Undirected graphing is an area of mathematics dealing with problems such as planning a delivery route to visit a number of shops while travelling the least distance, designing a communications network to link a number of towns, organising the flow of work in a factory, or allocating jobs for increased efficiency. Below are examples of undirected graphs or networks you may have come across: Route map for the Melbourne Metropolitan Tram Network H – O – H O= C = O (H 2 O) (CO 2 ) The chemical molecules for water (left) and carbon dioxide (right) Milliammeter Rectifier OA91 12 v, 24 w Lamp 1000 ohm Voltmeter + The Swiss mathematician Leonhard Euler (pronounced oyler; 1707–1783) developed much of the theory of undirected graphs in his work on topology and graph theory. AC 50 mA + DC + – An electrical circuit +
Chapter 21 Undirected graphs and networks 155 Vertices and edges The map at right shows the main roads linking a number of country towns in Victoria. We can see that there are three main roads leading into Traralgon, but only one joining Leongatha and Yarram. The map is an example of an undirected graph or network since there are no arrows showing a particular direction. The network consists of vertices (the towns) and edges (the roads). The degree of a vertex is defined as the number of edges leading to or from that vertex. Therefore the degree of the Traralgon vertex is 3 because there are 3 roads leading to or from there. The degree of the Yarram vertex is 2 because there are only 2 roads leading to or from there. The graph is considered to be a connected graph since it is possible to reach each vertex from any other one. A connected graph must have all vertices joined to at least one other vertex. There cannot be any isolated vertices. We use the term multiple edge if there is more than one edge linking two vertices. For example, in the figure at right there is a multiple edge between vertices A and C. The figure also contains a loop at vertex B. An edge which connects a vertex to itself is defined as a loop. When calculating the degree of a vertex, a loop counts as 2. Thus the degree of vertex B is 3; 1 for the edge connecting B to A plus 2 for the loop. Trafalgar Moe Morwell Traralgon Leongatha Yarram The figure is not a connected graph because not all vertices are connected to at least one other vertex. E is therefore an isolated vertex as it is not able to be reached from the other vertices. WORKED Example For the following graph, state: a the number of vertices b the number of edges c the degree of each vertex d whether the graph is connected. THINK WRITE a Count the number of vertices. Note: The vertices are the points labelled A, B, C, D and E. 1 b Count the number of edges. Note: The number of edges is found by counting the number of lines joining the vertices. Remember: A loop is counted as 2. c Count the degree of each vertex, that is, the number of edges leading to or from each vertex. Note: 1. C contains a loop, which is counted as 2 when totalling edges. 2. The degree of A is abbreviated to deg(A) etc. a The number of vertices is 5, that is, A, B, C, D and E. b The number of edges is 9. c Vertex A has 4 connections to other vertices, so deg(A) = 4. Vertex B has 3 connections to other vertices, so deg(B) = 3. Vertex C has 6 connections to other vertices, so deg(C) = 6. Vertex D has 3 connections to other vertices, so deg(D) = 3. Vertex E has 0 connections to other vertices, so deg(E) = 0. d Answer the question. d The graph is not connected, as vertex E has no edges leading to or from it; that is, it is isolated from each of the other vertices. A D A C D B E A B B C C E E D
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