School Year: 2016-2017 Discrete Mathematics

Subject: CST11 Math MCUS04 (Cycle 2 / Year 3) Solutions

Assignment Booklet 1Introduction to Fundamentals

and Key Concepts of Graph Theory

Name: _ Section: MCU504: _

1) A relation is defined by the graph below

A

c

a) Determine the order the graph above.lHE«e.,A-ee- 7 v€Rll' CE-S , \~~e:..~-e.../IH~ OP...J)E(2. \. So7

b) Are vertices B and D adjacent? Explain.

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c) Are edges BE and G (1) E adjacent? Explain.. YES B f-CA-vl s E- il BC-"A f\lD 'Cs- Cl) E 1/ S (-i-f-\ eG- 'j'1+E s AM E \l c.:.~,Sf C 5..) .

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d) Is it a connected graph? Explain.,

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Mr. Randimbiarison: CST11 Math - MCU504 - Discrete Mathematics - Graph Theory Page 1

2) Draw a complete graph with five vertices

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3) A relation IS defined by the graph 6. below (8 marks) ~ / s· 4 .l INLE EJ)&fS MlA5T t?~N ~ E...'D ~ D :rot< sA (t~ \lEJ:}, T G'-X

A To COrJ't0~c..1 \0 ',Tt~ Olf~'\I'€. i-l r'c.s ')

c

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e) Determine the order the graph above.I'H-£f$- ~ C; VE(L\('C.E S' , Tf~PGl~. l~ O?lD€:-Q I'~ 6

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f) Are vertices Band D adjacent? Explain.'!SSfB N~p)) ARf-A-t5:\ACtN\ )3~~lASE '"n+Ey ~Q5- I>;(2.G.C\~Y C.uI'fNE..C.~...b

\3 Y AN' \2-J>G-E:- (!3 - !)')g) Are edges C(3)D and F(2)B adjacent? Explain.

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h) Is it a connected graph? Explain. fJ 0 I 1"\ ,~ r0 0 T A

4) Write a conjecture about the relationship between the number of edges in a graph and the sum of itsdegrees. Draw three examples with the following requirements:

1st example: A connected graph of order 5 that contains one set of parallel edges2nd example: A connected graph of order 4 that contains two loops and one parallel edges3rd example: A connected graph of order 6 with exactly 5 edges and no parallel edges and no loops

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5) Draw a complete graph with six vertices

a) Evaluate the sum ofthe degrees ofthat graph.

D(S)

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FCS)

b) How many edges must be removed so that you can obtain a disconnected graph?

5~..s I"l1Ll5f-JR (t()l1)ve-d to is.oloft CLverh f ,wVlI'ch makes() I I' h lthQ. ~QfV\ ellS COYlnec ~.O-.

Mr. Randimbiarisont CST11 Math - MCUS04 - Discrete Mathematics - Graph Theory Page 4

6) For each of the diagrams below determine its order, number of edges, the sum of the degrees, and thenumber pairs of parallel edges.

a)

Order= S-Number of edges= q

b)

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Mr. Randimbiarison: CST11 Math - MCUS04 - Discrete Mathematics - Graph Theory Page 5

e)

10 2...-------.©

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g)

Order= :s-Number of edges= /2Sum of degrees = 5+ 5" + 6 + 4 + Lt ::- 2.Number of pairs of parallel edges: 2.

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f)

2(1)

Mr. Randimbiarison: CSTll Math - MCU504 - Discrete Mathematics - Graph Theory Page 6

7) Given the star graph R below, determine its order, number of edges, the sum of the degrees, and thenumber pairs of parallel edges.

a) A relation R is defined as"x+ 1 = y", where x and y are the values of thevertices of the star graph above. Which edge mustbe removed in the graph above to satisfy therelation R?

Answer: ..edJ e.. ~ .• 1)

b) Is R a connected graph? (Yes/No?). Explain.YES ( G.J2 c.£Vt.S{ tl,{ II'-et'-h~~ c:~,

8) Given the star graph Rbelow.A

G c

Ela) How many edges are needed to be drawn in

the four-pointed star graph above in order toobtain a complete graph? Justify your answer.

Order= 2Number of edges= 2

9) Graph game: "Parallel words"

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L

~_-I- ,NRe-draw the equivalent graph of the one on the leftwith the following constraints:

No edges can cross one anotherAdd a parallel edge to each existing edge.

List all the pairs of words that satisfy the relation Rfor every pair of parallel edges between each pair ofcurrently connected vertices. ijarnl2.(-eS ~

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