Maths Quest 11 Standard General Mathematics 1

Networks WorkSHEET 11.1 Name: ____________________ 1 From this graph, state the number of:

(a) vertices (b) edges.

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2 From this graph write: (a) the degree of each vertex (b) whether the graph is connected.

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3 This graph is unconnected.

Draw in the extra edge(s) to make the graph connected.

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Maths Quest 11 Standard General Mathematics 2

4 Draw the graph where the number of vertices is 4 with deg(A) = 3, deg(B) = 2, deg(C) = 3, and deg(D) = 2.

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5 Draw the graph with 4 vertices and deg(E) = 2, deg(F) = 1, deg(G) = 2, and deg(H) = 3.

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6 Redraw these graphs as planar graphs, if possible. (a)

(b)

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Maths Quest 11 Standard General Mathematics 3

7 Complete this table using planar graphs.

Write down Euler’s rule.

Vertices (V)

Regions (R)

Edges (E)

V + R – E

3 2 3 4 5 2 5 9 2 9 13 2 18 12 2

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8 A Eulerian path is not possible for this graph. Explain why?

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9 Which of these are Eulerian circuits? (a)

(b)

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Maths Quest 11 Standard General Mathematics 4

10 The graph below shows a number of roads in a suburban residential area. The council is organising a street-sweeping route so that the same road is not used twice. The street sweeping truck is to start and finish at the council depot, D.

(a) Explain why is it not possible to do this

without going down one street twice. (b) Which street has to be left out so that it is

possible to complete the task without going down the same street twice?

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Maths Quest 11 Standard General Mathematics 1

Networks WorkSHEET 11.2 Name: ____________________ 1 From the graph state:

(a) the number of vertices (b) the number of edges (c) the degree of each vertex (d) whether the graph is connected.

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2 Draw a graph that has 4 vertices and 8 edges.

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Maths Quest 11 Standard General Mathematics 2

3 Draw the graph that has 4 vertices such that deg(A) = 3, deg(B) = 5, deg(C) = 2, deg(D) = 2.

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4 Redraw this graph as a planar graph.

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Maths Quest 11 Standard General Mathematics 3

5 Complete the table of vertices (V), regions (R) and edges (E) for the 3 connected planar graphs.

Write down Euler’s rule for connected planar graphs.

Vertices (V)

Regions (R)

Edges (E)

V + R – E

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6 Use Euler’s rule to determine the number of edges in a connected planar graph with 40 vertices and 42 regions.

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Maths Quest 11 Standard General Mathematics 4

7 Explain why the Eulerian path for this graph could not be drawn.

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8 Explain why the Eulerian circuit for this graph is possible.

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9 A courier supplies, daily, the parcels to outlets A, B, C, D, E and F from a parcel distribution centre W. The distances (in km) along the roads are shown on the graph. Determine the shortest delivery route and the distance to each parcel outlet if the courier returns to the centre.

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Maths Quest 11 Standard General Mathematics 5

10 Find the shortest distance along the roads between the towns P and A as shown.

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