VALLIAMMAI ENGINEERING COLLEGE

Transcription

1 1 VALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur 0 20 DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING QUESTION BANK VII SEMESTER CS02- GRAPH THEORY AND APPLICATIONS Regulation 201 Academic Year Prepared by Mr. N. LEO BRIGHT TENNISSON, Assistant Professor/CSE Mrs.A. LALITHA, Assistant Professor/CSE

2 2 VALLIAMMAI ENGINEERING COLLEGE (A member of SRM Group of Institutions) SRM Nagar, Kattankulathur 020 DEPARTMENT OF COMPUTER SCIENCE AND ENGINEERING Year and Semester : IV/ VII Section : CSE 1 & 2 Subject Code : CS02 Subject Name : GRAPH THEORY AND APPLICATIONS Degree and Branch : BE-CSE Staff Incharge : Mr. N. Leo Bright Tennisson,Mrs. A. Lalitha UNIT 1 UNIT I INTRODUCTION Graphs Introduction Isomorphism Sub graphs Walks, Paths, Circuits Connectedness Components Euler graphs Hamiltonian paths and circuits Trees Properties of trees Distance and centers in tree Rooted and binary trees. PART A S. No. Question BTL Competence 1 Define fundamental numbers for a complete graph K BTL 1 Remember Draw induced sub graph for V={A,C,D} and show complement of the following graph 2 BTL Apply Show that a Hamiltonian path is a spanning tree. BTL Apply Define ring-sum of two graphs BTL 1 Remember

3 Give an example of an Euler graph which is arbitrarily traceable List any two properties of a graph with n 2 edges, where n is the number of vertices in the graph. In a directed graph, when do we say a vertex is isolated? Explain BTL 2 BTL 1 BTL Understand Remember Analyze Suppose G is a finite cycle-free connected graph with at least one edge. Describe to show that G has at least two vertices of BTL 1 Remember degree 1. 9 Explain isomorphism between graphs BTL Analyze Evaluate,Is the directed graph given below strongly connected? Why or Why not? 10 BTL Evaluate 11 Create a graph G with 1 edges and each vertex has degree. Determine the number of vertices for the graph G. Is the BTL Create graph G a simple graph? 12 Give two properties of binary tree. BTL 2 Understand For the following pair of graphs, determine whether or not the graphs are isomorphic. Give the justification for your answer. 1 BTL Evaluate 1 1 Describe the following with one example each : a) Sub-graph b) Spanning Sub-graph Show that Peterson graph has no Hamilton cycle but it has a Hamilton path BTL 2 BTL Understand Apply

4 1 Define walk, path and circuit in a graph BTL 1 Remember 1 Explain Hamilton Path and Cycle. BTL Analyze 1 Draw all simple graphs of one, two, three and four vertices and label it BTL 1 Remember 19 What is a Complete graph? Design a complete graph. BTL Create 20 Discuss Induced sub-graph with an example. BTL 2 Understand Part B 1 i) In a complete graph having odd number of vertices, how many edge disjoint Hamiltonian circuits exist? Explain ii) State the two theorems to check if a connected graph G is Eulerian. Explain with proof BTL Analyze 2 iii) Find a path of length 9 and a circuit of length in the Peterson graph. i) Illustrate the search algorithm than can be employed to find the components or blocks in a graph, with an example ii) Explain the following theorem with proof In a graph the number of the vertices with odd degree is even Give the proof for the following theorem BTL Analyze i) If a graph has exactly two vertices of odd degree, there must be a path joining these two vertices. ii) A connected graph is an Euler graph if and only if every vertex has even degree BTL 2 Understand iii) A connected graph is an Euler graph if and only if it can be decomposed into circuits i) Show that any two simple connected graphs with n vertices, all of degree two are isomorphic ii) Find v for the following graph or multigraphs G. a) G has nine edges and all vertices have degree b) G has ten edges with two vertices of degree and BTL 1 Remember all other of degree

5 i) Give the explanation to prove that any two simple connected graphs with n vertices, all of degree two are isomorphic ii) Give the explanation to prove that the following graphs G and H are not isomorphic BTL Analyze i) Prove that a tree with n vertices has (n-1) edges BTL1 Remember ii) State and prove Dirac s theorem Prove to show that graph G is disconnected if and only if its vertex set V can be partitioned into two nonempty subsets V1 and V2 such that there exists no edge in G whose one end vertex is in V1 and the other in V2 i) Determine whether the following graphs G and H are isomorphic. Give reason 1 BTL Apply BTL 2 Understand ii) Give the proof for the following theorem : A given connected graph G is an Euler graph if and only if all the vertices of G are of even degree i) Prove that a simple graph with n vertices and k 9 components cannot have more than ( n k)( n k 1) 2 edges BTL Evaluate ii) Which of the following simple graphs have a Hamilton Circuit or if no, a Hamilton Path?

6 i) Define isomorphism of graphs. Show that no two of the following three graphs a shown in figure are isomorphic. 10 BTL Apply ii) Define Euler circuit. Discuss Konigsberg bridge problem i) In the undirected graph 11 Find a) an a-a circuit of length BTL 2 Understand b) An a-a cycle of maximum length ii) Seven students of a class have lunch together at a circular table. Using Hamilton cycles, Predict the minimum number of days required for each of them to sit next to every member of the class. i) If G is an undirected graph with n vertices and e edges, 12 let min {deg( v)} and max {deg( v)}, then v v prove that 2( e/ n) ii) Define Hamilton cycle. How many edge-disoint BTL Create Hamilton cycles exist in the complete graph with seven vertices? Also, Design the graph to show these Hamilton cycles.

7 i) Let G=(V,E) be the undirected graph in the following figure. How many paths are there in G from a to h? How many of these paths have length? 1 BTL 1 Remember ii) Let G=(V,E) be an undirected graph, where v 2. If every induced subgraph of G is connected, can we identify the graph G? 1 i) Show that the maximum number of edges in a simple graph with n vertices is n(n-1)/2 ii) Define the following with one example each a) Infinite graph b) Hamiltonian path c) Component of a graph d) Euler graph e) Spanning subgraph BTL 1 Remember PART C 1 i) The Figure represents a floor plan with the doors between the rooms and the outside indicated. The real estate agent would like to tour the house, starting and ending outside, by going through each door exactly once. What is the fewest number of doors that should be added, and where should they be placed in order to make this tour possible?explain the reasons for your answer BTL Evaluate ii) Define closed-walk, open-walk, path and circuit. Take a graph of your choice and give an example to each one. 2 i) Nine members of committee have their dinner in round table. If no member sits near to the same neighbour more BTL Create

8 than once, how many days can this arrangement possible? Write all possible arrangements. ii)state four properties of a tree graph and prove them. Draw a graph with vertices representing the squares of a chessboard. Join these vertices appropriately by edges, each representing a move of the knight. You will see that in this graph every vertex Is of degree two, three, four, six or eight. How many vertices are of each type? Explain 1 BTL Evaluate Given a maze as shown in Fig, prepare a representation of this maze by means of a graph such that a vertex denotes either a corridor or a dead end (as numbered). An edge represents a possible path between two vertices. 1 BTL Create

9 9 UNIT 2 UNIT II TREES, CONNECTIVITY & PLANARITY Spanning trees Fundamental circuits Spanning trees in a weighted graph cut sets Properties of cut set All cut sets Fundamental circuits and cut sets Connectivity and separability Network flows 1-Isomorphism 2-Isomorphism Combinational and geometric graphs Planer graphs Different representation of a planer graph. PART A S. No. Question BTL Competence 1 Explain eccentricity of a graph BTL Analyze 2 Explain edge connectivity. BTL Analyze Define a spanning tree in a graph BTL1 Remember What are the different properties when a graph G with n vertices is called a tree? BTL1 Remember Can you construct a graph if you are given all its spanning trees? How? BTL Apply Show that non-empty intersection of two fundamental circuits in a graph is always a path BTL Apply Write down any four observations of the graph given below. BTL1 Remember Design all trees of n labeled and unlabeled vertices for n=1,2, BTL Create 9 A simple planar graph to which no edge can be added without destroying its planarity (while keeping the graph simple) is a maximal planar graph. Develop a proof to show that every region in a maximal planar graph is a triangle. BTL Create

10 10 10 Explain edge disjoint path and vertex disjoint path. BTL Analyze 11 Define 1-isomorphic and 2-isomorphic BTL1 Remember 12 Describe separable and non-separable graph. BTL2 Understand 1 Give an example of 1-isomorphic but not isomorphic graphs BTL2 Understand 1 State Max flow min cut theorem and prove it through an example BTL Evaluate Give the dual graph for the following planar graph. 1 BTL2 Understand 1 Discuss planar graph with one example. What are the applications of planar graph? BTL2 Understand 1 Show that the complete graph of vertices is self-dual. BTL Apply 1 What is the nullity of a complete of n vertices? BTL1 Remember 19 Complete matching is present in a tree Argue on this statement BTL Evaluate 20 In a tree, every vertex is a cut-vertex. Justify the claim. BTL1 Remember Part B 1 Give the proof for the following theorems i) Every tree has either one or two centers ii) In any tree, there are at least two pendant vertices BTL 2 Understand iii) Number of vertices in a binary tree is always odd iv) Number of pendent vertices in a binary tree is (n+1)/2 i) Explain the proof of the theorem 2 Two graphs are 2-isomorphic if and only if they have circuit correspondence BTL1 Remember ii) For the following graph, find the all maximal independent sets.

11 11 iii) If the distance d(x,y) between two vertices x and y in a graph is defined to be the length of the shortest path connecting them, then prove that the distance function is a metric i) Find an Euler Circuit for the graph shown in the following figure. 10 BTL 1 Remember ii) Predict all spanning trees of the graph in the following figure. i) Establish and prove the relation between vertex connectivity, edge connectivity and number of vertices and edges ii) Explain the proof of following theorem The largest number of edges in a planar graph is n-, where n is the number of vertices in, the graph i) State and prove theorems relating fundamental circuits and fundamental cut set w.r.t a spanning tree BTL Apply BTL Analyze

12 12 ii) Explain with proof that the number of labeled trees with n vertices is n n-2 i) Prove that the ring sum of two cut sets is either a third cut set or an edge disjoint union of two cut sets ii) Explain to show that the edge connectivity and vertex connectivity of the graph in the following figure are each equal to three. BTL Evaluate i) Prove that an Euler graph cannot have a cut-set with odd number of edges ii) Develop the algorithm to find cut vertices and bridges in a graph BTL Create i) Give the proof of the following theorem A spanning tree T of a given weighted connected graph G, is a shortest spanning tree of G if and only if there exists no other spanning tree at a distance of one from T whose weight is smaller than that of T BTL 2 Understand ii) Create the algorithm to find the shortest path from a given source vertex to any vertex in a graph with an example i) Explain the steps involved in testing for planarity of graphs 9 ii) Define graph isomorphism and characterize graphs possessing 1-isomorphism and 2-isomorphism BTL Analyze iii) Prove that every connected graph has at least one spanning tree i) State Kuratowski s theorem. Examine if the following graphs are planar or non-planar i) K, ii) K,. Give 10 reason. ii) Find the number of vertices, edges and regions for BTL 2 Understand the following planar graph and verify that Euler s Theorem for connected planar graphs is satisfied

13 1 11 Explain with proof that a graph is non-planar if and only if it contains a sub-graph homomorphic to K or K, i) Show that when any edge is removed from K, the resulting subgraph is planar. Is this true for the graph K, ii) Using the algorithm of Kruskal, find a shortest spanning tree in the following graph 1 BTL Analyze 12 BTL 1 Remember 1 i) Show that an Hamiltonian Path is a spanning tree ii) List all cutsets with respect to the vertex pair v2,v in the graph BTL Apply

14 1 1 i) In a network as shown below, during propagation of worms, protection strategy needs to be adopted against the attack. Find the optimal number of nodes where the defense strategy can be deployed using Boolean arithmetic. BTL1 Remember ii) Given a connected graph G, find the rank of a matrix that defines the graph within 2-isomorphism Part C 1 i) Show that starting from any spanning tree of a graph G, every other spanning tree of G can be obtained by successive cyclic interchanges. ii) Prove that every circuit has an even number of edges in common with a cut-set. i) Prove and explain the following: With respect to the given spanning tree T, a branch b i that 2 determines a fundamental cut-set S is contained in every fundamental circuit associated with the chords in S and in no others. ii) Show, by drawing the graphs, that two graphs with the same rank and the same nullity need not be 2-isomorphic State and explain minimum cut maximum flow theorem. Using it calculate the maximum flow between the nodes D and E in the graph (Fig). The number on a line represents the capacity. BTL Analyze BTL Evaluate 1 BTL Evaluate

15 1 i) Show that a vertex v in a connected graph G is a cutvertex if and only if there exists two vertices x and y in G such that every path between x and y passes through v ii) State Kuratowski's Theorem.Generalize and use it in order to prove the graph in Fig is non-planar BTL Create

16 1 UNIT UNIT III MATRICES, COLOURING AND DIRECTED GRAPH Chromatic number Chromatic partitioning Chromatic polynomial Matching Covering Four color problem Directed graphs Types of directed graphs Digraphs and binary relations Directed paths and connectedness Euler graphs. S. No. 1 2 PART A Question BTL Competence Given the adjacency matrix of a connected graph, how do you determine the diameter of the graph Explain how adjacency matrix representation of a graph helps in quickly checking if the graph is connected or not. BTL BTL Evaluate Analyze Define transitive closure of a diagraph with an example BTL 1 Remember Express the rank of a circuit matrix if e and n are respectively the number of edges and vertices of G. Suppose X is the adjacency matrix of a graph. What does X k denote? Define the two types of connectedness in digraphs. Give examples. Draw the graph represented by the given adjacency matrix Give the incidence matrix for the labeled simple graph. BTL 2 BTL 1 BTL 2 BTL 1 Understand Remember Understand Remember BTL 2 Understand

17 1 9 Find the Chromatic number of a complete graph of n vertices BTL 1 Remember 10 Show a graph with an even number of vertices but has no dimer covering. BTL Understand Show that the graph in the following figure has only one chromatic partition. What is it? Apply 11 BTL 12 Design two different non isomorphic graphs that have the same chromatic polynomial. BTL Create 1 Design all different simple digraphs with one, two and three vertices BTL Create 1 Is it possible for two non-isomorphic digraphs to have the same reachability matrix? BTL Analyze 1 Discuss about covering in a graph. BTL 2 Understand 1 What is a directed graph? Construct directed graph with vertices and 10 edges BTL Apply 1 Define equivalence graph BTL 1 Remember 1 List out various types of digraphs. Define Bipartite graph. BTL 1 Remember 19 Explain Euler digraph BTL Analyze 20 Define Chromatic number. Explain Chromatic Polynomial. BTL Evaluate Part B S. No. Question BTL Competence 1 i) Show that digraph representing the relation congruent mod on a set of finite integers 1-11 is an Equivalence graph BTL Evaluate

18 1 ii) Derive the chromatic polynomial for the given graph and use that to find information on chromatic number of the graph 2 i) Create an algorithm to find all cycles in a digraph using exhaustive search method and state the precautions to be taken. ii) Define a k-chromatic graph. Prove that every tree with two or more vertices is 2-chromatic. i) If A(G) is an incidence matrix of a connected graph G with n vertices then prove that rank of A(G) is n-1. ii)let a graph G is 2-chromatic then prove that it is bipartite.. iii)generate the circuit matrix for the following graph BTL 1 BTL Remember Create i) If the edges of a connected graph are arranged in the 10 same order for the columns of the incidence matrix A and the path matrix P(x,y), then prove that the product (mod 2) A.P T (x,y)= M, where the matrix M has 1 s in two rows x and y, and the rest of the n-2 rows are all zeros. BTL Analyze ii)explain the theorem The reduced incidence matrix of a tree is nonsingular. i) Define : 1. Adjacency matrix and 2. Incidence matrix of a graph with examples BTL Apply

19 19 ii) Show that a connected multi-graph has an Euler Circuit if and only if each of its vertices has an even degree i) Show that a graph with atleast one edge is 2 chromatic if and only if it has no circuits of odd length ii) Show that dmax is the maximum degree of the vertices in a graph G. BTL Apply i) What is an maximal independent set? State the procedure of finding all maximal independent sets. ii) What is an minimal dominating set? State the procedure of finding all minimal dominating sets. BTL 1 Remember i) Define mininal covering. Prove that a covering g of a graph is minimal if and only if g contains no paths of length three or more ii) Define the incidence matrix, of a graph G. Prove that the rank of an incidence matrix of a connected graph with n vertices is n-1. BTL 1 Remember 9 State and explain the Four-Color Problem 1 BTL Analyze 10 i) Explain isomorphic digraphs with example ii) List and explain various types of digraphs with neat BTL Analyze diagram 11 i) Prove that a graph of n vertices is a complete graph iff its chromatic polynomial is Pn(λ)= λ(λ-1)( λ-2)..( λ-n+1) BTL 1 Remember ii) Define a bipartite graph. Show that the complement of a bipartite graph need not to be a bipartite. 12 Give the proof for the following theorem 1 A diagraph G is an Euler digraph if and only if G is connected and is balanced [ie, d - (v)=d + (v) for every vertex v in G]. BTL 2 Understand 1 i) Explain the problem of partitioning in graph with example BTL 2 Understand ii) Give the proof for the following theorem

20 20 m-vertex graph is a tree if its chromatic polynomial is Pm (n) = n (n-1) (m-1) 1 Define the following and give one example to each : Complete Matching Minimal Covering Balanced Digraph Strongly Connected Digraph Fragment in a digraph. 1 BTL 2 Understand Part C i) Create the chromatic polynomial of the graph G in Fig using the theorem. Pn(λ) of G = Pn(λ) of G'+ Pn(λ) of G. 1 BTL Create ii) State and prove five-color theorem Describe chromatic number and chromatic polynomial of a graph. Deduce the chromatic number and chromatic polynomial of the following graphs 2 1 BTL Evaluate Design a geometric dual of the following planar graph and explain the procedure of construction. Also give the observations that can be made between the planar graph and its dual. 1 BTL Create

21 21 Prove and assess that a connected planar graph with n vertices and e edges has e-n+2 regions. 1 BTL Evaluate Unit UNIT IV PERMUTATIONS & COMBINATIONS Fundamental principles of counting - Permutations and combinations - Binomial theorem - combinations with repetition - Combinatorial numbers - Principle of inclusion and exclusion - Derangements - Arrangements with forbidden positions. Part A S. No. 1 Question BTL Competence In how many different ways can the letters of the word TRIANGLE can be arranged in such a way that the vowels BTL 1 Remember always come together? Explain 2 Show how many different bit strings are there of length seven BTL Apply Point out two basic principles of counting BTL Analyze. Give one example for rule of sum and rule of product. BTL 2 Understand. Define permutation. Give an example BTL 2 Understand. Find the number of permutations in the word COMPUTER if only five of the letters are used BTL 1 Remember

22 22. Find the number of arrangements of four letters in BALL BTL 1 Remember A committee including boys and girls is to be formed. from a group of 10 boys and 12 girls. How many different BTL Create committees can be formed from the group? 9. Show how many permutations are there for the eight letters a, c, f, g, i, t, w, x? BTL Apply 10. State Binomial theorem. BTL 1 Remember A student taking a history examination is directed to answer 11. any seven of 10 essay questions. There is no concern about order here. Explain how many ways the student can answer BTL Analyze the examination. 12. Express Binomial coefficient BTL 2 Understand 1. Find the Binomial coefficient of x y 2 BTL 1 Remember 1 A donut shop offers kinds of donuts. (Assume 2 of each kind). How many ways we can select 2 donuts. BTL Evaluate 1 Describe the principle of Inclusion and Exclusion BTL 2 Understand 1 If eight distinct dice are rolled, What is the probability that all six numbers appear? Explain BTL Evaluate 1 Find the number of de-arrangements of 1,2,, BTL 1 Remember 1 Show how many permutations of 1,2,,,,, is not dearrangement? BTL Apply 19. In how many ways can you invite at least one of your six friends to a dinner? Explain BTL Analyze 20. From consonants and vowels how many words can be created using consonants and 2 vowels? BTL Create Part B S. No. 1 Question BTL Competence i) Give the number of distinct permutations that can be formed from all the letters of each word BTL 2 Understand 1. RADAR 2. UNUSUAL

23 2 2 ii) If six people, designated as A, B,..,F are seated about a round table, how many different circular arrangements are possible, if arrangements are considered the same when one can be obtained from the other by rotation? iii) In how many can the symbols a, b, c, d, e, e, e, e, e be arranged so that no e is adjacent to another e i) Over the Internet, data are transmitted in structured block of bits called datagrams a) In how many ways can the letters in DATAGRAM be arranged? b) For the arrangements of part (a) how many have all three A s together? ii) Sixteen people are to be seated at two circular tables, one of which seats 10 while the other seats. How many different seating arrangements are possible BTL Analyze. i) Show how many positive integers n can be formed using the digits,,,,,, if we want n to exceed ? ii) A committee of 1 members,( 9 are women and are men) is to be seated at a circular table (with 1 seats).show In how many ways can the seats be assigned so that no two men are seated next to each other? i) Show How many numbers of arrangements can be made with the letters TALLAHASSEE having no adjacent A s? ii) In how many possible ways could a student answer a 10 question true-false test. iii) How many distinct four-digit integers can one make from the digits 1,,,, and? i) Show How many permutations of size can one produce with the letters m, r, a, f and t. BTL Apply BTL 1 Remember BTL 2 Understand

24 ii) A father, mother, 2 boys and girls are asked to line up for a photo Give the number of ways they can line up if a) No restriction b)parents stand together c)parents do not stand together d)all females stand together Determine the coefficient of x 9 y in the expansions of a) (x+y) 12 b) (x+2y) 12 and c)(2x-y) 12 Develop the details in the proof of the multinomial theorem i) Explain In how many ways can we distribute seven bananas and six oranges among four children so that each child receives at least one banana? iii) Explain with an example for the linear arrangements of the given n objects i) Rama has two dozen each of n different colored beads. If he can select 20 beads (with repetitions of colors allowed), in 20, 20 ways, calculate the value of n? ii) Determine the number of positive integers n where 1 n 100 is not divisible by or i) Explain in how many ways can the 2 letters of the alphabet be permuted so that none of the letters car, dog, pun or byte occurs? ii) Determine and Give the number of positive integers x where x and the sum of the digits in x equals 1 i) While at the racetrack, Ralph bets on each of the 10 horses in a race to come in according to how they are favored. In how many ways can they reach the finish line so that he loses all of his bets ii) Six married couples are to be seated at a circular table. In how many ways can they arrange themselves so that no wife sits next to her husband? 1 BTL Evaluate 1 BTL Create BTL Analyze BTL Apply BTL 2 Understand BTL 1 Remember

25 Let A = {1,2,,} and B={u,v,w,x,y,z}. How many oneto-one function f:a->b satisfy none of the following conditions. C1: f(1)= u or v, C2: f(2)=w, C:f()=w or x, C:f()=x,y or z i) In a box there are black pens, white pens, and red pens. In how many ways can 2 black pens, 2 white pens and 2 red pens can be chosen? ii) The Indian cricket team consists of 1 players. It includes 2 wicket keepers and bowlers. In how many ways can a cricket 11 be selected if we have 1 wicket keeper and bowlers? Evaluate a) C2 + C1 b) C2+ C1 c) C+ C2 d) 10 C2+ 10 C A committee of persons is to be formed from men and women. Explain in how many ways can this be done when ) At least 2 women are included ) At most 2 women are included 1 BTL 1 Remember BTL 1 Remember BTL Analyze Part C 1 There are five students in a group and their roll numbers are S1, S2, S, S, S and S. They are given with five assignments numbered 1 to. Each has to solve one assignment. How many ways the assignments can be distributed such that a student is not getting an assignment number same as his roll number? Analyze. Find the value of sum if the given program segment is executed. main () { int inc = 0, sum = 0; int i, j, k; for (i=1; i 10; i++) for (j=1; j I ; j++) for (k=1; k j; k++) { inc=inc+1; sum=sum+inc; BTL Analyze

26 2 2 } } i) In a survey of the chewing gum tastes of a group of baseball players, it was found that 22 liked juicy fruit, 2 liked spearmint, 9 like bubble-gum, 9 like both spearmint and juicy fruit, l liked juicy fruit and bubble gum, 20 liked spearmint and bubble gum, liked all three and liked none of these. Formulate how many baseball players were surveyed? ii) Using the principle of Inclusion and Exclusion find the number of prime numbers not exceeding 100. i) Explain and Show that if n and k are positive integers, then n 1 C n 1, k C n, k 1. Use this identity, k construct an inductive definition of the binomial coefficient. ii) How many integer solutions are possible for xl + x2 + x + x + x < 0 where xi -,1 i. i) A survey of 10 college students reveals that own cars, BTL BTL Create Evaluate 9 own bikes, 2 own motorcycles, own a car and a bike, 1 own a car and motorcycle, own a bike and a motorcycle and 2 own all the three. Deduce how many students own a bike and nothing else and how many students do not own any of the three? ii) Five professors P1,P2,P,P,P are to be made class advisor BTL Evaluate for the five sections C1,C2,C,C,C one professor for each section. P1 and P2 do not wish to become the class advisors for C1 or C2, P and P for C or C and P for C or C or C. In how many ways can the professors be assigned the work (without displacing any professor)?

27 2 Unit UNIT Generating functions - Partitions of integers - Exponential generating function Summation operator - Recurrence relations - First order and second order Nonhomogeneous recurrence relations - Method of generating functions. Part A S. No. Question BTL Competence 1 Give the generating function for a sequence of real numbers. BTL 2 Understand 2 Develop the sequence for the generating function (1+x) n. BTL Create What is the generating function for the sequence 1,1,1,1,1,,0,0,0,.where the first n+1 terms are 1. BTL 1 Remember Determine the coefficient of x 1 in f(x) = (x 2 +x +x +.) BTL Evaluate Develop all partitions of BTL Create What is the use of Ferrers graph? BTL 1 Remember Give the definition for generating function BTL 2 Understand Specify the exponential generating function for the sequence of real numbers. BTL Analyze 9 Examine the Maclaurin series expansion for e x BTL Apply 10 Why 1/(1-x) refers to summation operator. BTL 1 Remember 11 The number of virus affected files in a system is 00 (approximately) and this doubles every four hours. Using a recurrence relation, identify the number of virus affected files in the system after one day. BTL 1 Remember 12 Give the first-order linear homogeneous recurrence relation with constant coefficients. BTL 2 Understand 1 Solve the recurrence relation an= an-1 where n 1 and a2=9. BTL Apply 1 What is homogeneous and non-homogeneous relation? Explain BTL Analyze 1 State Lame s Theorem. BTL 1 Remember

28 2 1 Give the representation of Binet form. BTL 2 Understand 1 What is golden ratio? BTL 1 Remember 1 Explain the method of undetermined coefficients. BTL Evaluate 19 Solve the recurrence relation an+2 an+1 +an= -200, where n 0 and a0 =000, a1 =00. State second order linear homogeneous recurrence relation 20 with constant coefficients. BTL BTL Apply Analyze S. No. 1 2 Part B Question BTL Competence i) Solve the recurrence relation an= -an-1- an-2- an- given that a0=, a1=9 and a2=1. ii) Solve the recurrence relation S(n) = S(n-1)+2(n-1), with S(0)= S(1)=1, by finding its generating function i) A factory makes custom sports cars at an increasing rate. In the first month only one car is made, in the second month two cars are made, and so on, with n cars made in the n th month. 1) Give recurrence relation for the number of cars produced in the first n months by this factory. 2) How many cars are produced in the first year? ii) Find the generating function of Fibonacci sequence i) Find the generating functions for the following: a. 1,-1,1,-1,1-1,.. b. 1,0,1,0,1,0,.. ii) Determine the sequence for the following: a. f(x)=(2x-) b. f(x)=1/(-x) BTL Apply BTL 2 Understand BTL 1 Remember i) Find the coefficient of x 1 in each of the following: a. x (1-2x) 10 b. (x -x)/(1-x) BTL Analyze Analyze and solve the following i) Find the coefficient of x 0 in (x +x +x 9 +..)

29 29 ii) Find the value of n for the following: 2P(n,2)+0 = P(2n,2) Analyze and solve the following i) A company hires 11 new employees, each of whom is to be assigned to one of four subdivisions. Each subdivision will get at least one new employee. In how many ways can these assignments be made? ii) Determine and give the sequence by each of the following exponential generating functions: a. f(x)=e x b. f(x)=1/(1-x) BTL Analyze i) Find the exponential generating function for the number of ways to arrange n letters, n 0, selected from each of the following words. a. HAWAII b. ISOMORPHISM BTL 1 Remember ii) Find the exponential generating functions for the following: a. 1,-1,1,-1,1-1,.. b. 1,2,2 2,2,2,.. c. 0!,1!,2!,!, i) Analyze and Solve the recurrence relation for the following: a. an= 2(an-1 an-2), where n 2 and a0 =1, a1 =2. b. an= an-1 +an-2), where n 2 and a0 =1, a1 =. BTL 1 Remember ii) Define recurrence relation. Find a recurrence relation for the number of binary sequences of length n that have no consecutive 0 s i) Find the recurrence relation, with initial condition, that uniquely determines each of the following geometric progressions. a. 2,10, 0, 20,. b.,-1,,-12, BTL Analyze ii) Analyze and Solve the recurrence relation for fibonocci series. 9 i) Find the generating functions for the sequences a. 1,2,,,,..; b. 1,2,,,,, c. 0,1,0,0,0,.. d. 0,1,2,,, BTL 1 Remember

30 0 10 i) Analyze and Solve the recurrence relation 2an+=an+2 + 2an+1 - an,n 0, a0 =0, a1=1, a2=2. BTL Evaluate ii) Determine (1+ i) 10 i) Develop the solution for the recurrence relation an an-1 = ( n ), where n 1, a0 =2. 11 ii) The population of Mumbai city is,000,000 at the end of the year 201. The number of immigrants is 20000n at the end of year n. The population of the city increases at the rate of % per year. Use a recurrence relation to determine the population of the city at the end of 202 BTL Create 12 i) Solve the following recurrence relation by the method of generating functions: a. an+1 an = n, n 0 and a0 =1 b. an+1 an = n 2, n 0 and a0 =1 c. an+2 an+1 +2an = 0, n 0 and a0 =1, a1 = d. an+2 2an+1 +an = 2 n, n 0 and a0 =1, a1 =2 BTL Apply 1 i) Give the solution for the following recurrence relations a. an+1 an = 2n +, n 0, a0 =1 b. an+1 an = n 2 - n, n 0 and a0 = ii) Give the solution for the following recurrence relations a. an+1 2an =, n 0, a0 =1 b. an+1 2an = 2 n, n 0 and a0 =1 BTL 2 Understand 1 i) Give the solution for the following systems of recurrence relations. a. an+1 = -2an bn b. bn+1 = an + bn n 0, a0 =1 b0 =0 ii) Write short note on Summation operator BTL 2 Understand

31 1 1 2 Part C A person invests Rs. 0,000 at % interest compounded annually. Assess the following. i) Find the amount at the end of l st, 2 nd, rd year. ii) Write the general explicit formula. iii) How long will it take to double the investment? Use recurrence relation. i) Formulate the fractional de-composition and identify the sequence having the expression z 1 2z z2 generating function ii) Find the generating function of the sequence,, 9, 10, as a 1 BTL Evaluate BTL Create i) Evaluate the recurrence relation Yn+2-Yn+1+Yn=n+. ii) If an denotes the sum of the first n positive integers, find a recurrence relation for an and then solve it. BTL Evaluate Two cases of soft drinks, 2 bottles of one type and 2 bottles of another, are distributed among five surveyors who are conducting taste tests. Plan and solve the following. In how many ways can the bottles be distributed so that each surveyor gets at least two bottles of each type? And in how many ways can they be distributed so that each surveyor gets at least two bottles of one type and three of other type? Use generating function 1 BTL Create