Question BankUnit 1Introduction to Graph Theory

Objective: In this chapter some basic concepts of graph theory are introduced and we obtain some elementary results. Here we do an attempt to show how graphs can be used to represent almost any problem involving discrete arrangements of objects. And we also discuss the sub graphs, walks ,path, circuits and Euler lines, Hamiltonion paths and last but not the least the famous Konigsberg’s Bridge problem.Sl.No. Questions Marks

1) Define with an example : (i) Graph (ii) multigraph (iii) pseudograph (iv) simple graph (v) digraph (vi) regular graph (vii) complete graph (viii) bipartite Graph (ix) degree of vertex (x) adjacent vertices (xi) pendant vertex

10

2) Define with an example : (i) Subgraph of a graph (ii) spanning sub graph (iii) Complement of a graph (iv) Self complementary graph

8

3) Define with an example : (i) Path (ii) simple path (iii)circuit (iv) a connected graph.

8

4) Define with an example a) Union (b) intersection (c) Ring sum of two graphs

6

5) List all types of digraph. Given an example each and draw them. 6

6) Define (a) Decomposition of graph into two sub graphs (b) Deletion of a vertex from a graph

(c) Fusion of two vertices in a graph Give an example each

6

7) Draw a graph that has a Hamiltonian path that does not have a Hamiltonian circuit

6

8) Define (i) an Eulerian path and (ii) a Hamiltonian path, with an example each.

6

9) State and prove the necessary and sufficient condition for an undirected graph to possess an Eulerian path

7

10) State Konigsberg bridge problem . 7

11) Prove that there is always a Hamiltonian path in a directed complete graph.

7

12) Explain nearest neighbour method to obtain a Hamiltonian circuit in a graph.

7

13) Explain traveling salesperson problem. 714) Define

i) Connected graph ii) Spanning subgraph iii) Complement of a graph. Give one example for each.

6*

15) Explain with example graph isomorphism. Show that in a graph G the 7*

number of odd degree vertices is even.16) Write a note on “Konigsberg-bridge problem” 7*

17) If G=(V,E) be a connected undirected graph. What is the largest value for |V| = no. of vertices if |E| = no., of edges = 19 and deg(v)>= 4 for all vE V

6*

18) Prove that G has an Euler circuit if and only if G is connected and every vertex in G has an even degree for G an undirected without isolated vertices.

7*

19) Show that there is no graph with 12 vertices and 28 edges where the degree of each vertex is either 3 or 4.

6*

20) Determine |V|, given that G=( V,E ) is regular with 15 edges. 4*

Unit 2Planar GraphsObjective: In this chapter we shall subject the entire graph G to the following important question: is it possible to draw G in a plane without its edges crossing over? This question of planarity is of great significance from a theoretical point of view. In addition ,we study planarity and other related concepts that are useful in many practical situations. At last we will discuss the coloring problem.

Sl.No. Questions Marks

21) Define a Planar Graph with an example.State and prove Euler’s formula for a planar graph.

6

22) Prove that a graph K3.3 is a non-planar (ii) Show that the graph K5 is non planar

6*

23) Explain a Geometric dual of a graph. What is self – dual graph? 624) Prove that a graph has dual if it is planar. 6

25) Define : Dual of a planar graph. Give one example

26) Define (i) chromatic number (ii) chromatic partitioning of a graph. 6

27) Show that the maximum number of edges in a complete bipartite graph of n vertices is [n2/4]

6

28) Define : i) Planar graph ii) Complete Bipartite graph

6*

29) Show that in any connected planar graph with n vertices, e-edges and f-faces e-n+2 = f. (Euler’s formula)

7*

30) Define chromatic number and chromatic polynomial. Find the chromatic polynomial for the graph given below

7**

31) Prove that the complete graph on 5 vertices is non planar. 7*32) Let G be the following graph shown in fig1(a) Then 7*

i) How many connected subgraphs of G have 4 vertices and include a cycle? Also write these subgraphs.

33) Define 1) Isomorphism of graphs ii) Ring sum of graphs iii) circuit and cycle of a graph.

6*

34) If G =(V,E) is a loop free undirected graph with |v| =n>=3 and if |E|>= (n-1 )+2 2 then prove that G has a Hamiltonian cycle.

7*

35) Let G-(V,E) be a connected planar graph or multigraph with |V|=v and |E|=e. Let ‘r’ be the number of regions in the plane determined by a planar embedding of G. Then show that v-e+r=2.

7*

36) Define i) Dual of a planar graph ii) Chromatic number iii) Complete bipartite graph.

6*

37) Let G be an undirected graph with subgraphs G1,G2. If G=G1UG2=Kn. For some n€ Z+ , then prove the following with usual notations: \P(G,λ)=[P(G1,λ).P(G2, λ)]/ λn

7*

Unit 3Trees

Objective: In this chapter we focus on a special type of graph called a tree, probably the most important concept in graph theory, especially for those interested in applications of graphs. Here we define a tree and study its properties. We shall also point out some of its applications . Finally this chapter introduces you to the notion in the theory of graphs-the spanning trees.Sl.No. Questions Marks

38) Define with an example : (i) tree (ii) leaf (iii) branch node (iv) distance between two vertices (v) eccentricity of a graph (vi) Center of a tree (vii) directed tree (viii) rooted tree (ix) binary tree (x) Spanning tree (xi) minimal spanning tree.

10

39) Prove that (i) there is one and only one path between every pair of vertices in a tree T. (ii) if there is one and only one path between every pair of vertices in a graph G, G is a tree (iii) a tree with n vertices has (n-1) edges. (iv) a tree with two are more vertices has atleast two leaves (v) a connected graph with (n-1) edges in a tree (vi) a graph with n-1 edges that has no circuit is a tree.

10

40) Use the above algorithm & find a minimum spanning tree for a graph 8

of your own

41) What is a prefix code ? Use Huffman’s procedure for finding an optimal binary prefix code for the following weights assigning the code word for each weight (i) 5,7,8,15,35,40 (ii) 8,9,12,14,16,19 (iii) 3,4,5,6,12 (iv) 1,2,4,5,6,9,10,12 (v)

1,4,9,16,25,36,49,64,81,100

8

42) Define i) Treeii) Binary rooted treeiii) Prefix code. Give one example for each.

6*

43) Prove that a tree with n vertices has n-1 edges. 7*44) Obtain a prefix code to send the message ROAD IS GOOD using

labeled Binary Tree and hence encode the message7*

45) Which of the following sets represents the prefix code ? give reason.A = {000, 001, 01, 10 , 11}B = { 1, 00, 01, 000, 0001}

4*

46) Obtain a binary prefix code using the labeled binary tree. 4**

47) Find all the spanning trees of the graph 6*

48) Prove that tree with n vertices has n-1 edges. 6*

49) Find all the spanning trees of the graph 8(4+4)*

50) Construct an optimal prefix code for the symbols a,b,c,d,e,f,g,h,I,j that occur with respective frequencies 78,16,30,35,125,31,20,50,80,3.

7*

51) Define: a) cut set ,b) bridge, c) edge connectivity, d) vertex connectivity

7*

Unit 4Optimization and Matching

Objective: This chapter provides us with a sample of the ways in which graph theory enters into an area of mathematics called operation research. Here we examine a shortest path algorithm for weighted graphs and techniques for finding a minimal spanning trees. We also discuss problems dealing with the allocation of resources and matching in

Sl.No. Questions Marks52) Explain Kruskal’s algorithm for finding a minimum spanning tree of a 7*

graph.

53) Define a fundamental system of cut sets and a fundamental system of circuits.

7

54) Define a transport network and a flow in a transport network and explain with an example.

7

55) Use the labeling procedure to find a maximal flow in the following transport networks : Draw all the networks obtained after each step.

7

56) Define with an example each : (i) edge connectivity (ii) vertex connectivity (iii) separable graph (iv) 1- isomorphism graph (v) 2 – isomorphism graph (vi) circuit correspondence.

7*

57) Show that the edge connectivity and vertex connectivity of the graph are both equal to 3.

6

58) What is the edge connectivity of the complete graph of n vertices ? 6

59) Define with an example : (i) incidence matrix (ii) fundamental circuit matrix (iii) cut set matrix (iv) Path matrix (v) adjacency matrix.

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60) What is a (i) matching (ii)covering of a graph. 6

61) Show that the graph Has only one chromatic partition. What is it? 6

62) Define :i) Vertex Connectivity ii) Bridge iii) edge connectivityiv) Cut vertex with an example

6*

63) Prove that the maximum flow possible between two vertices a and b in a network is equal to the minimum of the capacities of all cut-sets with respect to a and b.

6*

64) Find the shortest spanning tree using Prim’s algorithm for the weighted graph below

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65) Explain Prim’s algorithm for finding a minimum spanning tree of a weighted graph.

10*

66) i) Define m-ary tree and complete m-ary tree.ii) How many internal vertices does a complete 5-ary tree with 817 leaves have?

7*

67) Obtain an optimal prefix code for the message FALL OF THE WALL. Indicate the code.

6*

68) Apply merge –sort to the list 1,7,4,11,5,-8,-3,-2,6,10,3. 7*69) Define matching and complete matching with examples 5*70) Define edge- connectivity and vertex –connectivity. Give an example

for each.5*

Unit 5Fundamental Principles of countingObjective: Enumeration, or counting may strike one as an obvious process that a student learns when first studying arithmetic. But then it seems very little attention is paid to further development in counting as the student turns to “ more difficult” areas in maths, such as algebra, geometry, trigonometry, and calculus. Consequently, this chapter provides some warning about seriousness and difficulty of “mere” counting.

Sl.No. Questions Marks

71) The chairs of an auditorium are to be labeled with a letter and a positive integer not exceeding 100 what is the largest number of chairs that can be labeled differently.

6

72) There are 32 microcomputers in a computer center. Each microcomputer has 24 ports. How many different ports to a microcomputer in the center are there?

6

73) In how many ways can one distribute 10 identical marbles among 6 distinct containers

6*

74) How many one-to-one functions are there from a set with m elements to one with n elements?

7

75) A student can choose a computer project from one of three lists. The three lists contain 23, 15, 19 possible projects respectively. How many possible projects are there to choose from?

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76) In how many ways can the letters in VISITING be arranged? For these arrangements how many have all three I’s together?

7

77) How many positive integers n can we form using the digits 3,4,4,5,5,6,7 if we want n to exceed 5,000,000?

8

78) In how many ways can 12 different books be distributed among 4 children so that (a) each. Child gets three books? (b) The two oldest children get 4 books each and the two youngest get two books each?

8

79) Determine the coefficient of (i) xyz 2 in (x+y+z)4

(ii) xyz 2 in (2x-y-z)4

(iii)xyz-2 in (x-2y+3z-1)4

7

80) Define the Catalan numbers. 681) Let m,n be positive integers with 1< n £ m. Prove that S(m+1, n)=

S(m, n-1)+ nS(m,n)6

82) State sum and product rule of counting. Give one example 6*83) How many nine letter words can be formed using letters of the word

DIFFICULT 7*

84) A question paper contains two parts A and B. Each contains 4 questions. How many different a student can answer 5 questions by selecting at least 2 questions from each part

7*

85) How many positive integers n can be formed using the digits 3, 4,4,5,5,6,7 if we want n to exceed 5,000,000?

6*

86) In how many ways can one arrange three 1’s and three -1’s so that all six partial sums are non negative.

6*

87) A message is made up of 12 different symbols and and is to be transmitted through a communication channel with 45 spaces between the symbols with atleast three spaces between each pair of consecutive symbols.In how many ways can transmitter send such a message..

6*

88) A woman has eleven close relatives and she wishes to invite five of them to dinner. In how many ways can she invite them in the following situations?i) There is no restriction on the choice.ii) Two particular persons will not attend separatelyiii) Two particular persons will not attend together

7*

89) i) Find the coefficient of xyz2 in the expansion of (2x-y-z)4 .ii) Find the number of integer solutions of x1+x2+x3+x4+x5=30, where x1>=2,x2>=3,x3>=4,x4>=2, x5>=0.

7*

90) Define i) Ramsay numbers ii) Stirling number of the second kind iii) The pigeonhole principle.

7*

Unit 6The Principles of Inclusion and exclusionObjective: With Venn diagrams to lead the way, in this chapter we will obtain a pattern called Principles of Inclusion and exclusion. Using this principle, we will restate each problem in terms of conditions and subsets. Using enumeration formulas on permutation and combination we solve some simpler problems and we allow the principle to manage our concern about over counting.

Sl.No. Questions Marks

91) How many positive integers not exceeding 1000 are divisible by 7 or 11?

6

92) Give a formula for the number of elements in the union of four sets. 693) For which nÎZ+ is f(n) odd? 694) List all the derangements of 1,2,3,4,5 where the first three numbers

are 1,2 and 3 in some order.7

95) How many permutations of 1,2,3,4,5,6,7 are not derangements? 796) Construct or describe a smallest chess board for which r 10 ¹ 0 797) Find the rook polynomial for the standard 8 X 8 chessboard. 798) State pigeon hole principle and generalized pigeon principle . show

that if any five numbers from 1-8 are chosen then two of them will add up to 9

7*

99) In how many ways can the 26 letters of the alphabet be permuted so that none of the patterns CAR,DOG,PUN or BYTE occurs?

7

100) In how many ways can one arrange the letters in CORRESPONDENTS so that : i) there are exactly two pairs of consecutive identical letters.ii) There are atleast three pairs of consecutive identical letters.

7*

101) In how many ways can the ineteger 1,2,3, …10 be arranged in a line so that no even integer is in its natural place.

6*

102) An apple , a banana, and an orange are to be distributed to four boys B1,B2,B3,B4. The boys B1 & B2 do not wish to have apple, the boys B3 does not want banana or mango and B4 returns orange. In how many ways the distribution can be made so that no boy is displeased.

7*

Unit 7Generating functions

Objective: In this chapter we continue our study of enumeration, introducing at this time the important concept of the generating functions. The power of generating function rests upon its ability not only to solve the kinds of problems we have considered so far but also to aid us in new situations where additional restrictions may be involved

Sl.No. Questions Marks103) How many integer solutions are there for the equation c1 + c2 + c3 +

c4 =25 if 0£ c i for all 1 £ I £ 4 ?8

104) Determine the coefficient of x15 in (x2 +x3 + x4 +…)4 6105) In how many ways can we select seven non consecutive integers

from {1,2,3,…50}?6

106) Show that the number of partitions of nÎZ+ where no sum m and is divisible by 4 equals the number of partitions of n where no even sum and is repeated.

7

107) Define the exponential generating function. 7108) Determine the sequence generated by each of the following

exponential generating functions.(i) f(x) = 3e3x

(ii) f(x) = ex + x2 (iii) f(x) = 1/ (1-x)

7

(iv) f(x) =e2x –3x3 + 5x2….+ 7x109) Find the exponential generating function for the sequence 0! 1!, 2!

……,7

110) Let f(x) be the generating function for the sequence a0, a1,a2,…. For what sequence is (1-x) f(x) the generating function?

7

111) Define generating functions and exponential generating functions. Give one example.

6*

112) Find the coefficient of x^18 in the product(x + x^2 + x^3 + x^4 + x^5 ) ( x^2 + x^3 + x^4 + -----)5

7*

113) Find the generating function for the sequence 0,2,6,12,20, 30, --- 7*114) Find the sequence corresponding to the generating function 3x^3 +

e^2x7*

115) Using the summation operator theory find a formula to express 0^2 + 1^2 + 2^2 + --- n^2 as a function of n

7*

116) Determine the generating function of the numeric function ar = 2^r if r is even = -2^r if r is odd

7*

117) Find a formula for the convolution of each of the following pairs of sequences:i) an = 1, 0≤n≤4 an = 0, n ³ 5 and bn=n, for all n€ Nii) ) an = bn = (-1)n, for all n € N

7*

118) In how many ways can we distribute 24 pencils to 4 children so that each child gets alleast 4 pencils but no more than nine.

6*

119) Find the number of ways in which 5 of the letters in “ENGINE” be arranged.

7*

Unit 8Recurrence relationObjective: In this chapter we study about how the recurrence relation can become as one of the tools for solving combinatorial problems. In these problems we analyze a given situation and then express the result an in terms of the results for certain smaller nonnegative integers. Once the recurrence relation is determined , we can solve for any value of an.

Sl.No. Questions Marks120) Find the general solution for each of the following recurrence

relations.(a) an+1 –1.5 an =0, n ³ 0(b) 4an-5an-1 = 0, n ³ 1(c) 2an-3an-1 =0, n ³ 1, a4 = 81

8

121) Suppose that the number of bacteria in a colony triples every hour (a) Set up a recurrence relation for the number of bacteria after n hours have elapsed (b) If 100 bacteria are use to begin a new colony, how many bacteria will be in the colony in 10 hours?

8

122) Find an explicit formula for the Fibonacci sequence. 7123) Prove that any two consecutive Fibonacci numbers are relatively

prime7

124) Solve the following recurrence relations(a) an = 5an-1 + 6an-2, n ³ 2, ao=1, a1=3.(b) an+2 + an = 0, n ³ 0 , ao=0, a1=3

7

(c) an + 2an-1 + 2an-2 ¹ 0, n ³ 2, ao=1, a1=3.125) Solve the following recurrence relations by the method of generating

functions(a) an+1- an = 3n, n ³ 0, , ao=1(b) an+1- an = n2 , n ³ 0, ao=1(c) an+2 - 3 an+1+ 2 an=0, n ³ 0, ao=1, a1=6

an+2- 2 an+1+ an= 2n , n ³ 0, ao=1, , a1=2

8

126) Solve the recurrence relation

F n+2 = F n+1 + F n given F0 = 0 and F1 = 1 and n>=0

6*

127) Find the general solution of s(k) – 3s(k-1) – 4s(k-2) = 4^k 7*128) Solve the non-homogeneous recurrence relation

an – 3an-1 = 57n where n>=1 and a0 =2. 7*

129) The number of virus affected files in a system is 1000 and this increases 250% every 2hours. Use a recurrence relation to determine the numbers of virus affected files after one day

7*

130) Find and sole the recurrence relation for the number of binary sequences of length that has no consecutive 0’s .

7*

131) Solve the recurrence relation :an+1 + 3 an+1+ 2 an=3n, n ³ 0,given , ao=0, a1=0

6*

132) Solve the recurrence relation by the method of generating functionsan+1 -an=n2, n ³ 0, given , ao=1

7*

133) Show that 97 is 25th prime number. 6*134) Show that any set of seven distinct integers includes two integers x

& y such that at lest one of x+y or x-y is divisible by 10.6*