Explain nearest neighbour method to obtain a Hamiltonian circuit in a g raph. ( Dec E xplain traveling salesperson problem. D efine * i ) Connected gr

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1 I ntroduction Q uestion Paper Unit 1 1 ) Define with an example : (i) Graph (ii) multigraph (iii) pseudograph (iv) simple graph (v) digraph (vi) regular graph (vii) complete graph (viii) b ipartite Graph (ix) degree vertex (x) adjacent vertices (xi) pendant v ertex ( June 09) 10 ) Define with an example : (i) Subgraph a graph (ii) spanning sub g raph (iii) Complement a graph (iv) Self complementary graph ( June 3 ) Define with an example : (i) Path (ii) simple path (iii)circuit (iv) a c onnected graph. ( Dec 4 ) D efine with an example a) U nion (b) intersectio n (c) Ring sum two graphs ( Dec 5 ) L ist all types digraph. Given an example each and draw them. ( June 0) ) D efine (a) Decomposition graph into two sub graphs ( b) Deletion a vertex from a graph ( c) Fusio n two vertices in a graph G ive an example each ( July 0) ) Define (i) an Eulerian path and (ii) a Hamiltonian path, with an example e ach. ( Jan 0) ) State and prove the necessary and sufficient condition for an undirected g raph to possess an Eulerian path 9 ) S tate Konigsberg bridge problem. Prove that there is always a Hamiltonian path in a directed complete g raph. ( June 1

2 Explain nearest neighbour method to obtain a Hamiltonian circuit in a g raph. ( Dec E xplain traveling salesperson problem. D efine * i ) Connected graph ii) Spanning subgraph i ii) Complement a graph. Give one e xample for each. ( July Explain with example graph isomorphism. Show that in a graph G the n umber odd degree vertices is even. ( Dec 0) 15) Write a note on Konigsberg- b ridge problem P lanar Graphs Unit 1 ) D efine a Planar Graph with an example. ( Dec 0) S tate and prove Euler s formula for a planar graph. ) Prove that a graph K3. 3 is a non-planar (ii) Show that the graph K5 is non p lanar ( July 0) * 3 ) E xplain a Geometric dual a graph. What is self d ual graph? 4 ) P rove that a graph has dual if it is planar. ( June 0) 5 ) D efine : D ual a planar graph. Give one example ( June 09) ) D efine (i) chromatic number (ii) chromatic partitioning a graph. ) Show that the maximum number edges in a complete bipartite graph n vertices is [n / 4] ( Dec 0) ) D efine : i ) P lanar graph i i) C omplete Bipartite graph ( June 9 ) S how that in any connected planar graph with n vertices, e-edges and f- faces e- n + = f. (Euler s formula) Define chromatic number and chromatic polynomial. Find the chromatic p olynomial for the graph given below ( July 0) * *

3 P rove tha t the complete graph on 5 vertices is non planar. ( July L et G be the following graph shown in fig1(a) Then i) How many connected subgraphs G have 4 vertices and include c ycle? Also write these subgraphs. ( Dec 09) a Define 1) Isomorphism graphs ii) Ring sum graphs iii) circuit and c ycle a graph. ( July I f G =(V,E) is a loop free undirected graph with v = n>=3 and if E >= ( n-1 )+ then prove that G has a Hamiltonian cycle. ( July 15) Let G- (V,E) be a connected planar graph or multigraph with V =v and E =e. Let r be the number regions in the plane determined by a planar embedding G. Then show that v- e +r=. 1) Define i) Dual a planar graph ii) Chromatic number iii) Complete b ipartite graph. ( Dec 09) 1) L et G be an undirected graph with subgraphs G1,G. If G=G1UG= Kn. For some n Z, then prove the following with usual notations: \ n P(G,λ)=[P(G1,λ).P(G, λ)]/ λ ( June * * U nit 3 T rees 1 ) D efine with an example : (i) tree (ii) leaf (iii) branch node (iv) distance between two vertices (v) eccentricity a graph (vi) Center a tree 10 3

4 (vii) directed tree (viii) rooted tree (ix) binary tree (x) Spanning tree (xi) m inimal spanning tree. ) P rove that (i) there is one and only one path between every pair vertices in a tree T. (ii) if there is one and only one path between every pair 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 e dges that has no circuit is a tree. 3 ) Use the above algorithm & find a minimum spanning tree for a graph y our own ( Dec- 0) 4 ) 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 e ach weight (i) 5,,,15,35,40 (ii),9,1,14,1,19 (iii) 3,4,5,,1 (iv) 1,,4,5,,9,10,1 (v) 1,4,9,1,5,3,49,4,1,100 ( Dec 09) 10 5 ) D efine * a. T ree b. B inary rooted tree c. P refix code. Give one example for each. ) Prove that a tree with n vertices has n- 1 edges. ( June 09) ) Obtain a prefix code to send the message ROAD IS GOOD using l abeled Binary Tree and hence encode the message ( June ) W hich the follow ing sets represents the prefix code? give reason. A = {000, 001, 01, 10, 11} 4 * B = { 1, 00, 01, 000, 0001} ( July 0) 9 ) O btain a binary prefix code using the labeled binary tree. 4 ** 4

5 F ind all the spanning trees the graph ( Dec 0) * Prove that tree with n vertices has n- 1 edges. * F ind all the spanning trees the graph ( July ( 4+4) * 5

6 Construct an optimal prefix code for the symbols a,b,c,d,e,f,g,h,i,j that o ccur with respective frequencies,1,30,35,15,31,0,50,0,3 (July 09) D efine: a) cut set,b) bridge, c) edge connectivity, d ) vertex connectivity U nit 4 O ptimization and Matching 1 ) Explain Kruskal s algorithm for finding a minimum spanning tree a g raph. ( Dec 0) ) D efine a fundamental system cut sets and a fundamental system c ircuits. ( June 0) 3 ) Define a transport network and a flow in a transport network and e xplain with an example. ( Dec 0) 4 ) Use the labeling procedure to find a maximal flow in the following t ransport networks : Draw all the networks obtained after each step. 5 ) D efine with an example each : (i) edge connectivity (ii) vertex connectivity (iii) separable graph (iv) 1- i somorphism graph (v) i somorphism graph (vi) circuit correspondence. ( June ) S how that the edge connectivity and vertex connectivity the graph are b oth equal to 3. ( July 0) ) W hat is the edge connectivity the complete graph n vertices? ) D efine with an example : (i) incidence matrix (ii) fundamental circuit m atrix (iii) cut set matrix (iv) Path matrix (v) adjacency matrix. 10

7 9 ) W hat is a (i) matching (ii)covering a graph. ( June 0) S how that the graph Has only one chromatic partition. What is it? D efine : a. V ertex Connec tivity ( July b. B ridge c. e dge connectivity d. C ut vertex with an example Prove that the maximum flow possible between two vertices a and b in a network is equal to the minimum the capacities all cut- sets with r espect to a and b. ( June 0) Find the shortest spanning tree using Prim s algorithm for the weighted g raph below ( July 0) * * * Explain Prim s algorithm for finding a minimum spanning tree a w eighted graph. ( June 0) 10 15) i) Define m-ary tree and complete m- a ry tree. ii) How many internal vertices does a complete 5- ary tree with 1 l eaves have? ( Dec 0) 1) O btain an optimal prefix code for the message FALL OF TH E WALL. *

8 I ndicate the code. 1) A pply merge sort to the list 1,,4,11,5, -, - 3, -,,10,3. 1) D efine matching and complete matching with examples ( June 5 * 19) Define edge- c onnectivity and vertex connectivity. Give an example 5 * f or each. ( June U nit 5 F undamental Principles counting 1 ) The chairs an auditorium are to be labeled with a letter and a positive integer not exceeding 100 what is the largest number chairs that can b e labeled differently. ( Dec 0) ) T here are 3 microcomputers in a computer center. Each microcomputer has 4 ports. How many different ports to a microcomputer in the center a re there? 3 ) I n how many ways can one distribute 10 identical marbles among * d istinct containers ( July 4 ) How many one-to- one functions are there from a set with m elements to o ne with n elements? ( Dec 0) 5 ) A student can choose a computer project from one three lists. The three lists contain 3, 15, 19 possible projects respectively. How many p ossible projects are there to choose from? ) I n how many ways can the letters in VISITING be arranged? For these a rrangements how many have all three I s together? ( Dec 0) ) How many positive integers n can we form using the digits 3,4,4,5,5,, i f we want n to exceed 5,000,000? ) I n how many ways can 1 different books be distributed among 4 children so that (a) each. Child gets three books? (b) The two oldest c hildren get 4 books each and the two youngest get two books each? 9 ) D etermine the coefficient (i) xyz in 4 (x+y+z) (ii) xyz in (x-y- z) 4 - (iii)xyz in (x-y+3z ) 4 ( Dec 0) D efine the Catalan numbers.

9 Let m,n be positive integers with 1< n m. Prove that S(m+1, n)= S(m, n- 1 )+ ns(m,n) ( July 0) S tate sum and product rule counting. Give one example ( July 0) * How many nine letter words can be formed using letters the word D IFFICULT 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 l east questions from each p art ( June 15) How many positive integers n can be formed using the digits 3, * 4,4,5,5,, if we want n to exceed 5,000,000? ( july 1) I n how many ways can one arrange three 1 s and three - 1 s so that all six * p artial sums are non negative. ( Dec 09) 1) A message is made up 1 different symbols and and is to be * t ransmitted through a communication channel with 45 spaces between the symbols with atleast three spaces between each pair consecutive s ymbols.in how many ways can transmitter send such a message.. 1) A woman has eleven close relatives and she wishes to invite five them t o dinner. In how many ways can she invite them in the following s ituations? ( July 0) i ) There is no restriction on the choice. i i) Two particular persons will not attend separately iii) Two particular persons wi ll not attend together 19) i) Find the coefficient xyz in the expansion (x-y- z) 4. i i) Find the number integer solutions x1+x+x3+x4+x5=30, w here x1>=,x>=3,x3>=4,x4>=, x5>=0. 0) D efine i) Ramsay numbers ii) Stirling number the second kind iii) The p igeonhole principle. ( July 0) U nit T he Principles Inclusion and exclusion 1 ) H ow many positive integers not exceeding 1000 are divisible by or 11? ) G ive a formula for the number elements in the union four sets. 3 ) For which n Z is ( n) odd? ( Dec 0) 9

10 4 ) L ist all the derangements 1,,3,4,5 where the first three numbers are 1, and 3 in some order. ( Dec 5 ) H ow many permutations 1,,3,4,5,, are not derangements? ) Construct or describe a smallest chess board for which r 10 0 ) F ind the rook po lynomial for the standard X chessboard. ( June ) State pigeon hole principle and generalized pigeon principle. show that if any five numbers from 1- are chosen then two them will add up to 9 ( June 9 ) In how many ways can the letters the alphabet be permuted so that n one the patterns CAR,DOG,PUN or BYTE occurs? I n how many ways can one arrange the letters in CORRESPONDENTS s o that : i) there are exactly two pairs consecutive identical letters. i i) There are atleast three pairs consecutive identical letters. In how many ways can the ineteger 1,,3, 10 be arranged in a line so * t hat no even integer is in its natural place. ( July An apple, a banana, and an orange are to be distributed to four boys B1,B,B3,B4. The boys B1 & B do not wish to have apple, the boys B3 d oes not want banana or mango and B4 returns orange. In how many w ays the distribution can be made so that no boy is displeased. ( June U nit G enerating functions 1 ) How many integer solutions are there for the equation c1 + c + c 3 + c 4 = 5 if 0 c i for all 1 I 4? ( dec 0) ) 15 Determine the coefficient x in (x + x + x 4 + ) 3 ) In how many ways can we select seven non consecutive integers from { 1,,3, 50}? ( Dec 0) 4 ) Show that the number partitions n Z where no sum m and is divisible by 4 equals the number partitions n where no even sum a nd is repeated. ( Dec 09) 5 ) D efine the exponential generating function. ) Determine the sequence generated by each the following exponential g enerating functions. ( i) 3x f(x) = 3e 10

11 x ( ii) f(x) = e x ( iii) f(x) = 1/ ( 1- x ) ( iv) x f(x) = e 3x 3 + 5x.+ x ( Dec ) F ind the exponential generating function for the sequence 0! 1!,!, ) Let f(x) be the generating function for the sequence a0, a1, a,. For what sequence is (1- x ) f(x) the generating function? ( June 0) 9 ) D efine generating functions and exponential generating functions. Give * o ne example. F ind the coefficient x^1 in the product ( x + x^ + x^3 + x^4 + x^5 ) ( x^ + x^3 + x^ ) 5 ( July F ind the generating function for the sequence 0,,,1,0, 30, --- u ne 09) F ind the sequence corresponding to the generating function 3x^3 + e^x U sing the summation operator theory find a formula to express ( July 0^ + 1^ + ^ n ^ as a function n D etermine the generatin g function the numeric function a r = ^r if r is even ( July = - ^r if r is odd 15) Find a formula for the convolution each the following pairs s equences: i ) an = 1, 0 n 4 a n = 0, n 5 n and b = n, for all n N ii) ) an = bn = (- 1) n, for all n N ( june 1) In how many ways can we distribute 4 pencils to 4 children so that each * c hild gets alleast 4 pencils but no more than nine. ( july 09) 1) Find the number ways in which 5 the letters in ENGINE be a rranged. ( june + U nit R ecurrence relation 1 ) F ind the general solution for each the following recurrence rela tions. ( a) a n an = 0, n 0 ( b) 4a n -5a n - 1 = 0, n 1 ( c) a n -3a n- 1 = 0, n 1, a 4 = 1 ( dec 0) ) S uppose that the number bacteria in a colony triples every hour (a) Set up a recurrence relation for the number bacteria after n hours h a ve elapsed 11

12 (b) If 100 bacteria are use to begin a new colony, how many bacteria w ill be in the colony i n 10 hours? ( dec 09) 3 ) F ind an explicit formula for the Fibonacci sequence. 4 ) P rove that any two c onsecutive Fibonacci numbers are relatively prime 5 ) S olve the following recurrence relations ( dec 0) ( a) a n = 5an-1 + an-, n, a o = 1, a1 = 3. ( b) a n+ + a n = 0, n 0, ao= 0, a1= 3 ( c) a n + an-1 + an- 0, n, a o = 1, a1 = 3. ) S olve the following recurrence relations by the method generating f unctions ( a) a n+ 1 - a n = 3, n 0,, ao= 1 ( b) a n+ 1 - a n = n, n 0, a o = 1 ( c) a n an+ 1+ an = 0, n 0, a o = 1, a1= a n+ - an+ 1+ a n =, n 0, a o = 1,, a1= ) S olve t he recurrence relation ( june 0) * F n + F = n + 1 F + n given F0 = 0 and F1 = 1 and n>= 0 ) F ind the general solution s(k) 3s( k- 1 ) 4s( k- ) = 4^k ( june 9 ) Solve the non- h omogeneous recurrence relation n a n 3a n- 1 = 5 w here n>=1 and a0 =. ( june 09) The number virus affected files in a system is 1000 and this increases 50% every hours. Use a recurrence relation to determine the numbers o f virus affected files after one day ( june F ind and sole t he recurrence relation for the number binary sequences o f length that has no consecutive 0 s. S olve the recurrence relation : ( june 0) * 3 + an= 3, n 0,given, ao= 0, a1= 0 a + 1 n + an+ 1 S olve the recurrence relation by the method generating functions a n + 1 -a n = n, n 0, given, ao= 1 ( july 09) S how that 9 is 5th prime number. ( dec * 15) S how that any set seven distinct integers includes two integers x & y * such that at lest one x+y or x- y is divisible by 10. ( june 0) 1

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CS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK

CS6702 GRAPH THEORY AND APPLICATIONS 2 MARKS QUESTIONS AND ANSWERS 1 UNIT I INTRODUCTION CS6702 GRAPH THEORY AND APPLICATIONS QUESTION BANK 1. Define Graph. 2. Define Simple graph. 3. Write few problems