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DEPARTM ENT OF APPLIED SCIENCES (M ATHEM ATICS)

Assignment No. 1

Title of Co urse: Discrete Structures

Course Code: AML4209

Topic Name: Sets, Relations and Functions

1. Which of the following relations are Symmetric?

(a) ‘is a sister’ on the set of all members of a family.

(b) ‘is a multiple of’ on the set of N.

(c) ‘is a devisor of’ on the set of Integers.

(d) ‘is perpendicular to’ on the set of lines on a plane.

2. In a survey of 500 television watchers 285 watch Football, 195 watch Hockey, 115 watch

Basketball, 45 watch Football and Basketball, 70 watch Football and Hockey, 50 watch Hockey and

Basketball, 50 do not watch any of the three games.

(a) How many watch all the three games.

(b) How many watch exactly one of the three games.

(c) How many watch Football but not Basketball and Hockey.

3. In a town of 10,000 families, it was found that 40% families buy newspaper A,20% buy newspaper

B, 10% buy newspaper C,5% buy newspaper A and B,3% buy newspaper B and C,4% buy

newspaper C and A and 2% buy all the three newspapers. Find the number of families which buy

newspaper

(a) A only

(b) B only

(c) None of A,B and C

4. 100 of 120 students at a college take at least one of the languages Hindi, English and French.65

study Hindi, 45 study English, 42 study French, 20 study English and Hindi,25 study Hindi and

French and 15 study English and French. Find the number of students studying all subjects. Also

find number of students studying exactly one subject.

5. Give an example of a relation which is reflexive but neither symmetric nor transitive.

6. Give an example of a relation which is symmetric but neither reflexive nor transitive.

7. If R and S be the following relations on A = { 1,2,3 }

R-1 = {(1,1),(1,2),(2,3),(3,1),(3,3)}, S = {(1,2),(1,3),(2,1),(3,3)}

Find (a) R ∩ S, R ∪ S, (b) RoS (c) S2 = SoS.

8. Let A= {1, 2, 3} and B={a,b,c,d}. In each case, state whether the given functions (if defined) is

Injective, surjective, bijective.

(a) f={(1,a),(2,d),(3,d)}

(b) g={(1,a),(1,b),(2,d),(3,c)}

(c) h={(1,a),(2,b)}

9. In a class of 40 students, 12 enrolled for both English and German. 22 enrolled for German. If the

Students of the class enrolled for at least one of the two subjects, then how many students enrolled

for only English and not German?

10. In a survey it was found that, the number of people that like only Pepsi, only Coke, both Coke and

Pepsi and neither of them are 2n, 3n, 69/n, 69/3n respectively. Find n?

11. Find the domain and range of f(x) = 2

x

x

12. Find the domain and range of f(x) =

2

2

2

2

x

x

13. If R is a relation defined as aRb, iff | a-b|>0, a ,b> 0 then the relation is

(a) Reflexive (b) symmetric (c) transitive (d) symmetric and transitive

14. If R is the relation in NXN defined by (a,b)R(c,d) iff a+d=b+c, show that R is an equivalence

Relation.

15. Among integer 1 to 300, how many of them are divisible by 3, nor by 5, nor by 7? How many of

Them are divisible by 3 but not by 5 nor by 7?

16. Let A={1,2,3,4,5,6} Define a relation R on A, R ={(x,y):x+y is a divisor of 24}

(i) Write elements of relation. (ii) Find the relational matrix MR. (iii) Discuss its properties.

17. The loudness of sound measured in decibels (dB) varies inversely as the square of the distance

Between the listener and the source of the sound. If the loudness of sound is 17.92 dB at a distance

of 10 ft from a stereo speaker, what is the decibel level 20 ft from the speaker?

18. A function is defined on the set of integers as follows

(i) Find the domain and range of the function.

(ii) Find the value of .

(iii)State whether is one-one or many one function.

19. Find the domain and range of the following functions:

(i) f( )=

(ii) f( )=

20. What is the sum of all integers from 1 to 100 that are multiples of 2 or 3?

21. A veterinarian surveys 26 of his patrons. He discovers that 14 have dogs, 10 have cats, and 5 have

fish. Four have dogs and cats, 3 have dogs and fish, and one has a cat and fish. If no one has all

three kinds of pets, how many patrons have none of these pets?

22. In a competition, a school awarded medals in different categories. 36 medals in dance, 12 medals

in dramatics and 18 medals in music. If these medals went to a total of 45 persons and only 4

persons got medals in all the three categories, how many received medals in exactly two of these

categories?

23. Given three sets P, Q and R such that:

P = {x: x is a natural number between 10 and 16},

Q = {y: y is a even number between 8 and 20} and

R = {7, 9, 11, 14, 18, 20}

(a) Find the difference of two sets P and Q

(b) Find Q – R

(c) Find R - P

(d) Find Q – P

24. Prove that (mod m) is an equivalence relation iff where .

25. Let A={1,2,3,4,5,6,7,8,9,10} .Find all min terms generated by B={4,5,6},C={7,8,9} and

D={1,2,3}.

26. Salad is made with combination of one or more eatables, how many different salads can be prepared

from onion, tomato, carrot, cabbage and cucumber.

27. If A={ +,-} and B={ 00,01,10,11}

(1) Find A X B

(2) How many elements of A4 and (A X B)3 have?

28. Consider A={1,2,3,4,5,6,7,8,9}. Let B1={5,6,7} , B2={2,4,5,9} and B3={3,4,5,6,8,9}. Find the

minsets generated by B1, B2, B3.

(a)Do these minsets form a partition of A.

(b) How many different subsets of A can you create using B1, B2, B3 with standard set operations.

29. Give example of relation which is:

(a) neither reflexive nor irreflexive

(b) both symmetric and anti-symmetric

(c) each reflexive, symmetric and transitive.

30. Let R be a relation on A whose corresponding directed graph is given below:

Determine the matrix on R.

31. Let and R and S be relations on A. Suppose the matrices of R and S are

, . Find

(a) (b) (c) (d)

32. Given Consider the following relation in A

(a) Draw its directed graph.

(b) Is R (i) Reflexive (ii) Symmetric (iii) Transitive (iv) anti-symmetric?

(c) Find .

33. Show that a function is bijective. If yes, then find its inverse.

34. Let f, g and h be functions from N to N, where N is the set of natural numbers so that

,

Determine fof, fog, gof, goh, hog, (fog)oh.

35. Is 1

1)(

x

xxf is invertible in its domain? If so find 1f .Also verify xxfof )(1 .

36. If A has 4 elements and B has 3 elements. Then find

a) Number of relation from A to B

b) Number of functions from A to B

c) Number of injective functions from A to B

37. If U={1,2,3,4,5,6}, A={1,2,3,4},B={3,4,5,6}.Find the bit string for the set A and B and use bit

string to find Ac and union and intersection of sets A and B.

38. How many integers between 1 and 60 that are not divisible by 2 nor by 3 and nor by 5.Also

Determine the number of integers divisible by 5, not by 2, not by 3.

39. How many elements in ( A x B) and (B x A) are common if n elements are common to A and B?

40. If R be a relation in the set of integers Z defined by R = {(x-y): x, y Є Z, (x-y) is multiple of 3}.

Show that it is an equivalence relation.

41. If f: N→N given by f(n) = 2n if n is even and f(n) = n if n is odd. Check whether f is one-one?

42. Determine whether the function f : R→R given below are 1-1 and onto R

i. f(x)=x+1

ii. f(x)=x2

iii. f(x)=|x|+x for all x Є R

iv. f(x)=x3

43. The function f : R→R given by f(x)=2x+1 for 0 ≤ x < 2

=x-2 for 2 ≤ x < 5

Find domain and range of f. Whether the function is 1-1 or many one?

44. If A is a collection of sets. Check whether the relation of “Subset” on A is Partial Order Relation or

not?

45. Let A = {1, 2, 3, 4, 5, 6}. Define a relation R on A and R = {(x, y): x +y is a divisor of 24}.

i. Find the relational Matrix M of R.

ii. Compute M2 and use M and M2 whether or not R is transitive.

46. Let R be the relation from S = {1, 2, 3, 4} to T = {a, b, c} with Boolean Matrix

010

001

100

101

i. Check RoR-1 is a symmetric relation on S.

ii. Are the relation R-1oR and RoR-1 equivalence relation.

47. A computer 350 applications from computer graduates for a job planning a new line of web

Servers. Suppose 220 of these people majored in computer science, 147 majored in business and

51 majored both in computer science and business. How many of these majored neither in computer

science or business.

48. Use Ceiling function, How many bytes are required to encode n bits of data for the following

values of n:

a) 7

b) 17

c) 1001

d) 28800

49. If R is the relation on N x N defined by (a, b) R (c, d) if and only if ad = bc, show that R is an

equivalence relation.

50. Let S = {red, blue, green, yellow}. Determine whether or not each of the following is a partition of

S.

i. P1={{red}, {blue, green}}

ii. P2={{red, blue, green, yellow}}

iii. P3={{S}}

DEPARTM ENT OF APPLIED SCIENCES (M ATHEM ATICS)

Assignment No. 2

Title of Co urse: Discrete Structures

Course Code: AML4209

Topic Name: Recurrence Relations, P & C , Generating functions and Pigeonhole Principle

1. Many horror stories involve zombie outbreaks. Suppose a zombie outbreak were to occur on

Planet Earth. 13 zombies are unleashed and when they bite humans, the human immediately

turn into zombies. The zombie population triples every hour. Write a recurrence relation for 𝑍𝑛, the zombie population n hour after the outbreak and hence solve it.

2. For a photo, the staff at a company have been arranged such that there are 10 people in the front

row and each row has 7 more people than in the row in front of it. Write a recursive formula,𝑎𝑛,

that completely defines the number of people in the 𝑛 row. Write an explicit formula, 𝑎𝑛, for the

number of people in the nth row.

3. Solve the recurrence relation: 𝑎𝑟 − 4𝑎𝑟−1 + 4𝑎𝑟−2 = (𝑟 + 1)2𝑟 . 4. Solve the following recurrence relation 𝑡𝑛 = 𝑡𝑛−1 + 𝑛, 𝑡1 = 4. and indicate if it is a linear

homogeneous relation or not. If yes, give its degree and if not justify your answer.

5. What is the minimum number of students required in a class to be sure that at least 6 will receive

the same grade if there are five possible grades A, B,C, D and F?

6. It is given that White tiger population in Orissa (India) is 30 at time 𝑛 = 0 and 32 at time 𝑛 = 1.

Also the increase from time 𝑛 − 1 to n is twice the increase from time 𝑛 − 2 to time 𝑛 − 1. Write

the recurrence relation for growth rate of tiger and then solve it.

7. The number of bacteria in a colony doubles every hour. If a colony begins with 5 bacteria, how

many bacteria will be there after n hours? Find a recurrence relation to represent the same and

hence solve it.

8. Obtain and solve a recurrence relation for Fibonacci sequence. 9. The snake shapes below are made using blocks, each with a side length of 1 unit. The perimeter

of each snake shape can be found by counting the sides of the blocks around the outside of the

shape. Construct a recurrence relation for perimeter of this problem and hence solve the

recurrence relation.

10. By using pigeonhole principle, show that if any five numbers from 1 to 8 are chosen, then two

of them will add up to 9.

11. A single section of fencing is made from four logs. Two sections use seven logs. Examples of

one, two and three-sections fences are shown below.

Construct a recurrence relation for the above problem and hence solve it.

12. There are 15 points in a plane, out of which 6 are collinear

(i) How many straight lines can be formed by joining them?

(ii) How many triangles can be formed by joining them?

13. State Tower of Hanoi puzzle and hence frame and solve recurrence relation for the problem.

14. Obtain partial fraction decompositions and identify the sequence having the expression as a

generating function 230111

296

zz

z

.

15. If U(n) = 2 . 3n, n ≥ 0, then G(U, z) is an infinite geometric series 2+2.3z+2.(3z)2+……. With

a = z, r=3z.

16. Find the generating function and sequence of the recurrence relation an+3an-1 = 0 with a0 = 7.

Write the Generating function of the sequence Sn = 4.3n+ 5(-1)n + 9.

17. Find the least number of cables required to connect 100 computers to 20 printers to guarantee

that 20 computers can directly access 20 different printers. Justify your answer.

18. A student must take 5 classes from 3 areas of study. Various classes are offered in each

discipline but the student cannot take more than two classes in any given area. Using the

pigeonhole principle, show that the student will take at least one class in each area.

19. Show that if 9 colors are used to paint 1000 houses, atleast 112 houses will be of the same

color.

20. Show that if any 4 numbers 1 to 6 are chosen, then two of them will add to 7.

21. Find generating function of sequence 1, 2/3, 3/9, 4/27, .……

22. Find the value of 12+22+32+………+n2 using generating function.

23. If S(k) – 4S(k-1)+3S(k-2) = k2, then find the generating function of S, G(S,z) ?

24. Solve S(k) – 8S(k-1)+21S(k-2)-18S(k-3)=0 for n>=3 using generating function upto partial

fractions.

25. Among a set of 5 black balls and 3 red balls, how many selections of 5 balls can be made such

that at least 3 of them are black balls.

26. How many 4 digit numbers that are divisible by 10 can be formed from the numbers 3, 5, 7, 8,

9, 0 such that no number repeats?

27. In how many different ways can the letters of the word 'CORPORATION' be arranged so that

the vowels always come together?

28. A box contains 4 different black balls, 3 different red balls and 5 different blue balls. In how

many ways can the balls be selected if every selection must have at least 1 black ball and one

red ball?

29. In how many ways can 10 engineers and 4 doctors be seated at a round table if no two doctors

sit together?

30. A company has 11 software engineers and 7 civil engineers. In how many ways can they be

seated in a row so that all the civil engineers do not sit together?

31. Solve the recurrence relation 𝑎𝑟 − 7𝑎𝑟−1 + 10𝑎𝑟−2 = 0 by the method of generating functions

with the initial conditions𝑎0 = 1, 𝑎1 = 2.

32. Find the generating function for the following sequences. In each case try to simplify the

answer.

a) 1,1,1,1,1,1,0,0,0,0….

b) 1, 1, 1, 1, 1…..

c) 1,3,3,1,0,0,0,0,…..

d) 𝐶02005, 𝐶1

2005, 𝐶22005, … … . . , 𝐶2005

2005, 0,0,0,0, … …

DEPARTMENT OF APPLIED SCIENCES (M ATHEM ATICS)

Assignment No. 3 Title of Co urse: Discrete Structures Course Code: AML4209 Topic Name: Logics , Lattice n Boolean

1. If you play golf during a thunderstorm, you’ll get hit by lightning. You didn’t get hit by

lightning. Therefore, you didn’t play golf during a thunderstorm

2. If I am literate, then I can read and write. I can read but I can’t write. Thus, I am not literate.

3. Simplify the given expression

A(A + B) + (B + AA)(A + B)

4. The card you picked is either a Jack, a Queen or a King. The card you picked is not a

Jack. Therefore, the card you picked is either a Queen or a King.

5. Consider the following conditional statement:

If the flood destroy my house or the fires destroy my house, then my insurance company will

pay me.

Write the converse, inverse and contrapositive of the statement.

6. Prove that D48 is a lattice.

7. Prove the Boolean expression:

(𝑥 ∨ 𝑦) ∧ (𝑥 ∨ ~𝑦) ∧ (~𝑥 ∨ 𝑧) = 𝑥 ∧ 𝑧 8. Find the Boolean expression for the following system:

9. Write Boolean expression for switching Circuits

10. Simplify the given expression

�̅�. 𝐵. 𝐶 + 𝐴. 𝐵.̅ 𝐶 + 𝐴. 𝐵. 𝐶̅ + 𝐴. 𝐵. 𝐶 11. Is D42 is complemented lattice?

12. Which of the following subsets of Lattice L represented by the Hasse diagram below are

sublattices of L? Justify your answer.

L1={0,a,b,1}, L2={0,a,e,1}, L3={a,c,d,1}, L4={0,c,d,1} ?

13. Is {S18, ʌ, v,ˈ, /, 18} Boolean algebra? Explain the reason.

14. Are the following lattices or not?Explain.

15. Find maximal, minimal, greatest(Last/Unit) element and least(First/Zero) element in

the following posets:

16. Prove or disprove the following equivalence,

~ p qp~ qq~ p

17. The following hasse diagrams represent the various focal points in Chandigarh. Help a person

to design his tour by finding maximal, minimal, greatest(Last/Unit) element and

least(First/Zero) element in the following posets:

18. A student has used following commands in his program as an arguments.

Rhombus R is a square or a parallelogram.

Rhombus R is a parallelogram.

Therefore, Rhombus R is not a Square.

Determine whether the argument is valid. Justify your answer.

19. Draw Hasse Diagram of Relation R on A.

Where A={ 1,2,3,4,5} and

20. Find the atoms of D30, D42, Boolean Algebra B={(0,0),(1,0),(0,1),(1,1)}.

21. Write the Boolean expression for the following circuits

22. Consider the following open propositions over the universe

U 4,2, 0, 1, 3, 5, 6, 8, 10

𝑃(𝑥): 𝑥 ≥ 4 , 𝑄(𝑥): 𝑥2 = 25,R(x) :is a multiple of 2.

Find the truth values of

(i) 𝑃(𝑥)⋀𝑅(𝑥) (ii) 𝑃(𝑥)⋀[∼ 𝑄(𝑥)]

23. Find the converse , inverse and contrapositive of the statement “If 8x-2=14 then x=2”

24. Simplify the given expression

AB + (AC)’ + AB’C(AB+ C)

1 2

3

4 5

c b

d

g e

a

b

c

d b

a

c e

25. Consider the following argument and determine whether it is valid.“Either I will get good

marks or I will not graduate. If I did not graduate I will go to Canada. I get good marks. Thus, I

would not go to Canada.”

26. Translate the following into symbolic form and test the validity of the argument:

“If 6 is even then 2 does not divide 7. Either 5 is not prime or 2 divides 7. But 5 is prime,

therefore, 6 is odd.

27. From the following formulae, find out tautology, contingency and contradiction.

a) 𝐴𝐴 ∧ (𝐴 ∨ 𝐵)

b) (𝑝 ∧∼ 𝑞) ∨ (∼ 𝑝 ∧ 𝑞)

c) ∼ (𝑝 ∨ 𝑞) ∨ (∼ 𝑝 ∨∼ 𝑞) 28. Let L be a bounded distributive Lattice, then show that if compliment of an element in L exists

then it is unique.

29. What are the different types of quantifiers? Explain in brief. Show that

(∃𝑥) (𝑃(𝑥) ∧ 𝑄(𝑥)) ⟹ (∃𝑥) 𝑃(𝑥) ∧ (∃𝑥)𝑄(𝑥)

30. Output 1’s appear in the truth table for these input conditions : ABCD = 0001; ABCD = 0110 ;

ABCD = 1110; Find the Sum of Products Equation.

31. Design a circuit (Given 3 inputs) which gives a high input when there is even number of low

inputs.

32. a) Write out the operations tables for {B22, -, , }

b) Draw the Hasse Diagram for {B22, -, , }

c) Find the atoms of this Boolean Algebra.

33. Let A={0,1} and f: A2 A , be defined by f(x,y) = (x )''()'()'() yxvyxvyxvy i.e. f is in

complete DN form. Determine all values of f(x,y) where x,y A .

34. Construct the circuit given by the following function f = (a )'()' bavb .

35. Construct a Boolean function of three variables 𝑝, 𝑞 and 𝑟 that has an out put when exactly two

of the 𝑝, 𝑞, 𝑟 having values 0 and an out put 0 in all other cases.

36. Draw logic circuit diagrams for the following

i. (𝐴 + 𝐵)(𝐵 + 𝐶)(𝐶′ + 𝐴′)

ii. 𝑥𝑦𝑧 + 𝑥 ′𝑦𝑧 ′

37. Draw a 3 input circuit which gives a high input when there is even number of low inputs.

38. Represent the Boolean expression (𝑥 + 𝑦)(𝑦 + 𝑧)(𝑥 + 𝑧) with the help of NOR gate only.

39. Simplify: (𝐴 + 𝐶)(𝐴𝐷 + 𝐴𝐷̅̅ ̅̅ ) + 𝐴𝐶 + 𝐶

40. In a Boolean algebra 𝐵 which of the following is not an atom

a) {2}

b) {18}

c) {2,18}

d) All of them

41. In D24, let the relation / (divide) be a partial ordering on D24.

a) Find all lower bounds of 8 and 12

b) Find glb of 8 and 12

c) Find least element of D24.

d) Determine lub of 8 and 12

e) Draw Hasse diagram for D24 with divide(/).

42. Which of the following posets figures are lattice, explain?

e

h g e

d

c f

c c

b

a

(a) (b) (c) (d) (e)

43. Check whether D6, D10, D15,{5,10,15,30} are sub lattices of D30. Also draw Hasse diagram for

D30.

44. Show that the lattice (L3≤ 3) of 3 tuples of 0 and 1 is complemented.

45. Consider the lattice L

a3

a) Which of the following are sublattices of L?

L1={0,b,c,1}, L2={0,b,e,1}, L3={0,b,a3,d,1},

b) Find complements ,if exits for b,c,a3

c) Is L is distributive

d) Is L is complemented lattice

46. Propositions p, q, r and s are defined as follows:

p is "I shall finish my Coursework Assignment"

q is "I shall work for forty hours this week"

r is "I shall pass Maths"

s is "I like Maths"

Write each sentence in symbols:

(a) I shall not finish my Coursework Assignment.

(b) I don’t like Maths, but I shall finish my Coursework Assignment.

(c) If I finish my Coursework Assignment, I shall pass Maths.

(d) I shall pass Maths only if I work for forty hours this week and finish my Coursework

Assignment.

47. The following predicates are defined:

friend is "… is a friend of mine"

wealthy is "… is wealthy"

clever is "… is clever"

boring is "… is boring"

Write each of the following propositions using predicate notation:

a. Sue is wealthy and clever.

b. Jane is wealthy but not clever.

c. Both Mark and Elaine are friends of mine.

b d

g

d

a b

a

a

d c b

d f

c b

0

d e

1

c b

a

d e

f

d. If Peter is a friend of mine, then he is not boring.

e. If Jimmy is wealthy and not boring, then he is a friend of mine.

48. Prove that

i) qqpp )(

ii) rprqqp )()( are tautologies

49. Formalize the following sentences:

a. “If David comes to the party then Bruno and Carlo come too”

b. “Carlo comes to the party only if Angelo and Bruno do not come”

c. “David comes to the party if and only if Carlo comes and Angelo doesn’t come”

d. “If David comes to the party, then, if Carlo doesn’t come then Angelo comes”

e. “Carlo comes to the party provided that David doesn’t come, but, if David comes, then

Bruno doesn’t come”

50. For the statement S: qp , explain the concept of inverse, converse and contrapositive of S with

the help of truth table.

`

AssignmentNo.4

TitleofCourse:Discrete Structures

CourseCode:AML4209

TopicName:Graph Theory

1. Determine the number of edges in a graph with 6 nodes, 2 of degree 4 and 4 of degree 2. Draw two

such graphs.

2. Determine the diameter of the graph .

V2

V3

V1

V6

V7

V8

V4 V5

3. Determine whether it is possible to construct a graph with 12 edges such that 2 of the nodes have

degree 3 and the remaining nodes have degree 4.

4. Find the number of edges in a graph with 6 vertices, 4 of degree 2 and 2 of degree 3.

5. A graph G has 21 edges, 3 vertices of degree 4 and other vertices are of degree 3. Find the number

of vertices in G.

6. Find the degree of each vertex of directed graph

7. Consider the graph:

A B C

D E F

Find

(i) All simple paths from A to F

(ii) All trials from A to F

(iii) D(A, F)

(iv) Size of G

(v) All cycle which starts from A

(vi) All cycle from B

8. Find eccentricity of all vertices, diameter of graph and radius of graph.

9. Use adjacency matrix to represent the graphs shown below

10. Draw the undirected graph represented by the adjacency matrix for the following

11. Draw the directed graph associated with the following matrices.

12. Represent the following graphs by using incidence matrix.

a)

b)

13. There are two different chemical molecules with formula 𝐶4𝐻10. Draw the graphs corresponding to

these formulas.

14. Draw the graphs corresponding to the chemical molecules of methane and propane.

15. Show that the following graphs are isomorphic by using adjacency matrices.

16. Given the floor plan of a house. Each room is connected to every room with which it has a

common wall and to the outside along each wall. Is it possible to begin in a room or

outside and take a walk through each door exactly once? If yes then find a Eulerian circuit

or Path.

17. Write down all possible path from v1 to v8, circuits of G and trails of length 3.

18. Indicate the nature of the following sequence of vertices from the figure given below.

19. Find all the circuits from the following figure

20. Draw the graphs 𝐶6 and 𝑊6.

21. Draw a cycle graph which is isomorphic to its complement.

22. Which of the following graphs are traversable?

23. Are the two graphs isomorphic ?

24. Give an example graphs for each of these :

i) A planar graph has 5 vertices and 3 faces. How many edges does it have?

ii) A planar graphs has 7 edges and 5 faces. How many vertices does it have?

25. Prove that the complete graph K4 is planar.

26. Prove these two graphs are isomorphic

27. Consider multigraphs G in figure,which of them are cycle-free.

D

B

A

C

E

(1)

N P

L

(2)

A B C

D E

(3)

M

28. Draw a multi-graph with four vertices and which is 5-regular graph.

29. Verify whether the graph is Pseudo graph or not ?

30. Draw the planar graph of K5.

31. Verify whether the graphs are complement to each other ?

32. Find the cut vertices and cut edges for the following graphs:

a)

b)

c)

33. Prove that the sum of degree of regions of a map is equal to twice the number of edges.

34. Give an example of a graph which is bipartite iff it contains no odd cycle.

35. Draw a Graph in which no edge is a cut edge

36. Construct a graph in which every edge is a cut edge

37. How many regions must a planar graph define if it has 11edges and 7 nodes?

38. Find the number of regions of:

a)

b)

39. Find a Spanning subgraph of:

a)

b)

40. (a) Construct a graph which is Hamiltionain but not Eulerian.

(b) Construct a graph which is Eulerian but not Hamiltonian.

(c) Construct a graph which is Eulerian as well as Hamiltonian

(d) Check whether the following given graph is Hamiltonian or Eulerian :

41. Every complete bipartite graph nmK , is Hamiltonian if

(a) nm (b) nm

(c) nm (d) nm

42. Every complete graph nK is Hamiltonian for

(a) 2n (b) 3n

(c) 1n (d) 1n

43. An undirected graph possesses an eulerian circuit iff (a) it is not connected and all its vertices are of odd degree (b) it is connected and all its vertices are of even degree (c) it is connected and all its vertices are of odd degree (d) it is not connected and all its vertices are of even degree 44. Which of the following graph contain no Eulerian circuit ?

(a) (b)

(c) (d)

45. Which of the following is False?

(a)A complete graph has a Hamiltonian circuit (b) If vertex of a graph is not of even degree, then it

does not have Euler circuit

(c) A Eulerian circuit for a graph G is a sequence of adjacent vertices and distinct edges in which every edge of G appears exactly. (d) A Eulerian circuit for a graph G is a sequence of adjacent vertices and distinct edges in which every vertex of G appears exactly.