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QUESTION 11 Determine the adjacency matrix that represents this graph. o1 H o oo o N — — O O Y — O —~ O o —~— O O n H (B) N — — O RIOI oo —~ — O w (A) H o oo o N oo o AN O Ao <o oo O wn = (D) o o — o N — — O — Y — O —~ O o —~ — O Ox n H= ©)
QUESTION 13 Which of the following is a planar graph with 5 vertices and 4 faces? ) L j ) % ©) J\ (D) ‘§‘
QUESTION 16 (3 marks) Use the graph to identify whether each of the following is a cycle an open walk an open trail or a closed trail. D a) AFCFB [1 mark] b) AFCEFBA [1 mark] c) ABCDEFA [1 mark]
QUESTION 2 How many faces does this planar graph have? (A) 3 (B) 4 C) 5 (D) 7
QUESTION 8 A basketball competition has six teams that have completed three rounds of competition as shown. - - - [ | Bls | — | - | v | v | = Mo | v | v | = v = s — | v = v e | - - | v | = S| v = | = | v The graph to represent this information has (A) 6 edges. (B) 9 edges. (C) 15 edges. (D) 18 edges.
QUESTION 2 The total number of vertices in this graph is (A) (B) ©) (D) ~N O WD W
QUESTION 6 This semi-Eulerian graph can be changed to an Eulerian graph by (A) adding a loop to vertex B. (B) removing the loop at vertex A. (C) adding an edge between vertices A and D. (D) removing the edge between vertices B and C.
QUESTION 21 (5 marks) The paths connecting various landmarks in a park are shown. D P Key B Bus stop C Coffee shop D Duck pond P Playground ¢ R Rose garden R W Water feature B a) Identify one cycle that passes through the rose garden and the playground. [1 mark] b) Identify whether the graph is Eulerian or semi-Eulerian. Justify your response. [2 marks] c) Construct an adjacency matrix from the graph using the vertex order listed in the key. [2 marks]
QUESTION 4 (5 marks) The table shows the current road length (in kilometres) between six towns. v [ vern [ o | e [ R | A T [ 6 s | [ % e | e | s e | e e | s w Cwew | T s e The government plans to build a direct road between Manon and Farra. Use a network diagram to determine the length of the direct road if it is to be 4 km shorter than the length of the current shortest road route between Manon and Farra.
QUESTION 13 Four athletes Eoin (E) Fedir (F) Gede (G) and Hana (H) compete in three events: javelin (J) long jump (L) and sprints (S). Which bipartite graph represents this information? “ok J ok J F L F L G S G S H H (C) (D) oo o- - mraoantt n ron n
QUESTION 20 (4 marks) Graphs 1| and 2 are shown. | 2 a) Show that Euler’s formula applies to graph 1. b) What feature in the drawing should be changed to represent graph 2 as planar? c) Redraw graph 2 to represent it as a planar graph. [2 marks] [1 mark] [1 mark]
QUESTION 23 (5 marks) A network graph is shown. ec a) State the degree of vertex E. [1 mark] b) State the number of edges joining D and E. [1 mark] c) Construct an adjacency matrix from the graph with the vertices in alphabetical order. [3 marks]
QUESTION 5 (7 marks) At 9:00 am a security guard begins their patrol of the eight work sites represented in the network diagram starting and ending at site A. They drive at 40 km/h on the roads between sites and check every site once for 15 minutes. The length (km) of each road corresponds to the terms of the arithmetic sequence ¢ = ¢ + 2(n — 1) where ¢ = 1. Determine the earliest possible time the security guard can finish their patrol and identify the route they must follow. B ty C
QUESTION 3 The graph below has six vertices. The vertex with the smallest degree is A) 1 (B) 2 ©C) 3 (D) 4
QUESTION 9 The edges J R M P P Q on the network diagram above make (A) apath. (B) acycle. (C) aclosed trail. (D) an open walk.
QUESTION 14 Identify the planar graph that has four faces. (A) g (B) j (C) B (D) ;
QUESTION 18 Which adjacency matrix below represents the same information that is given in the network diagram? T & To (B) To (A) oo o = UOllO _ o —~ © @ — N — o o o n = =2 woI > © © — © Do -~ o o _ o © © - N — o o o n = =2 woI To (D) To ©) b o = = O UOlOl - © — @ — N — o o o N == woxL J S e —~ 8 o Do o © = _ o © —~ O N — ©o o o n == WOL]
QUESTION 25 (3 marks) The connected weighted graph below shows the time taken in minutes and seconds to walk between places in a park. Rose garden 1 min 55 sec 1 min 50 sec Butterfly house 3 min 10 sec 2 min 30 sec 2 min 20 sec 4 min 55 sec Tlayground Garden shed 2 min 20 sec Cafe a) Identify which two places are joined by a network bridge. [1 mark] b) Calculate the earliest time that a gardener could arrive at the butterfly house if they leave the garden shed at 8:55 am. [2 marks]
QUESTION 2 (4 marks) A cable company is calculating the cost of laying cable between five houses (A B C D and E). The unit cost of laying the cable is $2.50 per metre. The shortest distances (in metres) between the houses are shown below. a) Construct a network diagram to represent the distances between the houses in the space provided below. [1 mark] b) Determine the minimum cost to connect all of the houses. [3 marks]
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