BHARATHIDASAN ENGINEERING COLLEGE NATTARAMPALLI Department of Science and Humanities CS6702-GRAPH THEORY AND APPLICATION

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BHARATHIDASAN ENGINEERING COLLEGE NATTARAMPALLI 635 854 Department of Science and Humanities DEGREE/BRANCH : B.E. CSE YEAR/ SEMESTER : IV/VII. CS6702-GRAPH THEORY AND APPLICATION 1. Define graph.

1 BHARATHIDASAN ENGINEERING COLLEGE NATTARAMPALLI Department of Science and Humanities DEGREE/BRANCH : B.E. CSE YEAR/ SEMESTER : IV/VII. CS6702-GRAPH THEORY AND APPLICATION 1. Define graph. UNIT-I (INTRODUCTION) PART-A 2. Define Loop. 3. Define Pseudo graph. 4. Define complete graph. 5. Define Regular graph. 6. State Fundamental theorem. 7. Show that the number of odd vertices is always even. 8. Define Bipartite graph. 9. Give an example for not isomorphic with 8 vertices. 10. Define Path. 11. Define Euler graphs. 12. Define Euler trail. 13. Define Hamiltonian path. 14. Define Tree. 15. Give all the non isomorphic graphs on 4 vertices. 16. Show that a Hamiltonian path is a spanning tree. 17. Define spanning tree. 18. Define Minimally connected graph. 19. Define Metric. 20. Define Rooted tree. 21. Define Binary tree. 22. What are the maximum and minimum height of a binary tree with 15 vertices. 23. Can you draw a tree with 3 vertices having degree 1,2,2.

2 24. Define walk, path and circuit in a graph (N/D-16). 25. What is meant by eccentricity? (N/D-16). 26. Define Euler graph. Show that an Euler graph is connected except for any isolated vertices the graph may have. (A/M-17) 27. Can there be a path longer than a Hamiltonian path (if any) in a simple, connected, undirected graph? Why? (A/M-17) PART-B 1. A simple graph with n vertices and k components can have atmost (n-k)(n-k+1)/2 edges. 2. A given connected graph G is an Euler graph iff all vertices of G are of even degree. 3. Let G be a simple connected graph. If there exists non adjacent vertices u and v such that d(u)+d(v) n, then G is Hamiltonian iff G+uv is Hamiltonian 4. Show that the number of vertices of odd degree in a graph is always even 5. State and prove Fundamental theorem. 6. A connected graph G contains Euler trail iff G contains atmost 2 vertices of odd degree. 7. State and prove Dirac theorem. 8. Prove that, there is one and only one path between every pair of vertices in a tree T. 9. Show that In a graph G, there is one and only one path between every pair of vertices, then G is a tree. 10. Show that a tree with n -vertices has n-1 edges 11. Show that In any tree, there are at least two pendant vertices. 12. Show that the maximum number of edges in a simple graph with n vertices is n(n 1)/2.(6) (N/D- 16) 13. Prove that if a graph has exactly two vertices of odd degree, there must be path joining these two vertices.(5) (N/D-16) 14. Prove that any two simple connected graphs with n vertices, all of degree two, are isomorphic. (5) (N/D-16) 15. Mention some of the properties of tree. (5) (N/D-16) 16. Prove that in any tree, there are atleast two pendant vertices. (5) (N/D-16) 17. Show that a Hamiltonian path is a spanning tree. (6) (N/D-16) 18. Define the following terms: Walk, Euler path, Hamiltonian path, Subgraph, Circuit, Complete graph. (6)(A/M-17) 19. From the given graph draw the following: (i)walk of length 6,(ii) Is this an Euler graph? give reasons,(iii)is there a Hamiltonian path for this graph? Give reasons,(iv) Find atleast two complete subgraphs. (10) (A/M-17) 20. List any five properties of trees. (6) (A/M-17)

3 21. Define eccentricity of a vertex V in a tree T and give an example tree and its eccentricity from the root. (10) (A/M-17) 1. Define spanning tree. 2. What is Branches and chords? 3. Define Rank and nullity. UNIT-II (TREES, CONNECTIVITY AND PLANARITY) 4. Show that a hamiltonian path is a spanning tree. 5. What is weighted graph? PART-A 6. State Max-Flow and Min cut theorem and give an example. 7. Define 2- isomorphism. 8. Define 1-isomorphism. 9. What is edge connectivity? 10. What is vertex connectivity? 11. Give an example of 1-isomorphic but not isomorphic graph. 12. Define cut set. 13. Define Fundamental cut-set. Give an example. 14. Define Separability. 15. Define planar Graph. 16. Define non-planar graph. 17. Define 1-isomorphism and 2-isomorphism.(N/D-16). 18. What are the applications of planar graph? (N/D-16). 19. Define planar Graphs. (A/M-17). 20. Identify two spanning trees for the following graph : (A/M-17) A 1 B 10 C E 9 D 2 3 PART-B

4 1) Every connected graph has atleast one spanning tree.( 8 mark). 2) Show that a path is its own spanning tree.( 8 mark) 3) Show that a Hamilton path is a spanning tree.( 8 mark). 4) A graph is a tree iff it is minimally connected.( 8 mark). 5) Define fundamental cut-sets with an example.( 6 mark). 6) The ring sum of any two cut-sets in a graph is either a third cut-sets or an edge-disjoint union of cut-sets.( 10 mark). 7) A vertex v in a connected graph G is a cut vertex iff there exists two vertices x and y in G such that every path between x and y passes through v.( 8 mark). 8) State and prove Max-flow and Min-cut theorem.( 8 mark). 9. Explain Max-flow and Min-cut theorem. (10) (N/D-16) 10. Explain about fundamental cut-sets and fundamental circuit in a graph. (6) (N/D-16) 11. Prove that every connected graph has atleast one spanning tree. (6) (N/D-16) 12. Prove that the graphs K 5 and K 3,3 are non planar. (10) (N/D-16) 13. Define spanning tree and give an example. (A/M-17) 14. State the Eulers formula relating the number of vertices, edges and faces of a planar connected graph. Give two conditions for testing for planarity of a given graph. Give a sample graph that is planar and another that is non-planar. (16) (A/M-17) UNIT III ( MATRICES COLOURING AND DIRECTED GRAPH ) PART A 1. Define minimal dominating set and maximal independent set. (N/D-16) 2. Find the chromatic number of a complete graph of n vertices. (N/D-16) 3. Does the following graph have a maximal matching? Give reason.(a/m-17) 1 a 2 b 3 c d e 4. Draw K 8 and K 9 and show that thickness of K 8 is 2 while thickness of K 9 is 3.(A/M-17) 5. In a directed graph,when do we say a vertex is isolated. 6. Define Euler digraph.

5 7. What is the upper limit of the chromatic number? 8. Define chromatic number. 9. Define directed graph and undirected graph. 10. Define proper colouring. 11. Find all maximal independent set of the toll graph. a b c d 12. Define maximal independent set. 13. Determine the chromatic number of the following graph. a f b e c 14. Define the term matching and maximal matching of the graph. 15. Explain digraph and give an example. PART-B 1. Derive the chromatic polynomial for the given graph and use that to find information on chromatic number of the graph? 2. Prove that an Euler graph cannot have a cut-set with odd number of edges? 3. Show that if a bi-partite graph has any circuits, they must all be of even lengths? 4. Prove that in any digraph the sum of the in degree of al vertices is equal to the of their out-degree and this sum is equal to the number of edges in the digraph? 5. State and prove five colour theorem? 6. A graph G is 2-chromatic if and only if G is bipartite? 7. State and prove stable marriage theorem? 8. Every tree with two or more vertices in 2-chromatic? 9. A graph with at least one edge in 2-chromatic if it has no circuits of odd length? 10. Prove that Every tree with two or more vertices is 2-chromatic. (5) (N/D-16) 11. Prove that a graph with n vertices is complete graph iff its chromatic polynomial is Pn(λ)= λ (λ -1) (λ -2).( λ -n+1). (6) (N/D-16) 12. Prove that a covering g of a graph is minimal iff g contains no paths of length three or more. (5) (N/D-16) 13. Explain Euler digraph. (10) (N/D-16) 14. Discuss about some types of digraph with suitable example. (6) (N/D-16) 15. If G is a tree with n vertices then show that its chromatic polynomial is p n (λ) = λ(λ 1) n 1. (N/D-16)

6 16. Describe the steps to find adjacency matrix and incidence matrix for a directed graph with a simple example. (16) (A/M-17) 17. Write a note on chromatic polynomials and their applications. (16) (A/M-17) UNIT IV (PERMUTATION AND COMBINATIONS) PART A 1. In how many different ways can the letters of the word LEADING be arranged in such a way that the vowels always come together?(n/d-16) 2. A committee including 3 boys and 4 girls is to be formed from a group of 10 boys and 12 girls. How many different committees can be formed from the group? (N/D-16) 3. State the rule of sum, the first principle of counting.(a/m-17) 4. Use venn diagram to represent the following scenario: If S: a set, C 1 = condition 1 and C 2 - condition 2 satisfied by some elements of S, indicate on the diagram S, N(C 1 ), N( C 2 ) N(C 1, C 2 ) and N(C 1, C 2 ). (A/M-17) 5. Find the co-efficient of x 9 y 3 in the expansion of (x + 2y) Find the number of derangements of 1,2,3,4. 7. Find the number of non-negative integer solutions of the equation x 1 + x 2 + x 3 + x 4 + x 5 = For the positive integer 1,2,3, n there are derangements where 1,2,3,4,5, appear in the first five positions. What is the value of n? 9. Define permutations and give formula permutations. 10. Find r if 6P r = Find the value of p(4,3). 12. Find n if P(n,4) = 42P(n,2) 13. For non-negative integers n and r, if n+1>r, prove that P(n + 1, r) = ( n+1 n+1 r ) P(n,r). 14. Ten students are participating in a race. In how many ways can the first four prizes be won? 15. Find the number of permutations of the letters of the word MASSASAUGA. In how many of these, all four A s are together?.how many of them begin with S? 16. Find how many three letter words can be formed out of the word Triangle. 17. Find the number of arrangements of all the letters in TALLAHASSEE. How many of these arrangements have no adjacent A s? 18. Determine the co-efficient of x 5 y 2 in the expansion of (x + y) Find the coefficient of x 0 in the expansion of (3x 2 2 x ) Determine the coefficient of xyz 2 in (w + x + y + z) Find the coefficient of w 2 x 2 y 2 z 2 in the expansion of (2w x + 3y + z 2) Find the sums of all coefficients in the expansion of (i) (x + y + z) 10 (ii) (2s 3t + 5u + 6v 11w + 3x + 2y) Find the number of positive integer solutions of x 1 + x 2 + x 3 = Find the number of non-negative integer solution of the in equality x 1 + x 2 + x x 6 < Compute ϕ(n) for n= Find ϕ(n) for n equal to How many derangements of 1,2,3,4, Evaluate d 6, d Give the formula for combination.

7 PART-B 1). Determine the number of positive integers n such that 1 n 100 and n is not divisible by 2, 3 or 5. (8) 2). Find the number of non-negative integer solution of the equationx 1 + x 2 + x 3 + x 4 = 18 under the condition x i 7 for i=1, 2, 3, 4. (8) 3).Determine the number of positive integer s n, 1 n 2000, that are (a) not divisible by 2, 3 or 5 (b) not divisible by 2, 3, 5 or 7. (16) 4). Find the rook polynomial for the board shown below (shaded part). (8) ). Fruits A 1, A 2, A 3 and A 4 are to be distributed to four boys B 1, B 2, B 3 and B 4. The boys B 1 and B 2 do not wish to have A 1, the boy B 3 does not want A 2 or A 3 and B 4 refuses A 4. In how many ways the distribution can be made so that no boy is displaced? 8) 6). Find the number of integer solutions of x 1 + x 2 + x 3 + x 4 + x 5 = 30, Where x 1 2, x 2 3, x 3 4, x 4 2, x 5 0. (8) 7). Compute φ(n), Where n= (8) 8). Five professors P 1, P 2, P 3, P 4 and P 5 are to be made class advisor for five sections C 1, C 2, C 3, C 4 and C 5 one professor for each sections. P 1, and P 2 do not wish to become the class advisors for C 1 or C 2, P 3 and P 4 for C 4 or C 5, and P 5 for C 3 or C 4 or C 5. In how many ways can the professors be assigned the work (without displacing any professor)? (8) 9). Four officers P 1, P 2, P 3, P 4 who arrive late for the dinner party find that only one chair at each of five tables T 1, T 2, T 3, T 4 and T 5 is vacant. P 1 will not sit T 1 or T 2, P 2 will not sit at T 2, P 3 will not sit at T 3 or T 4 and P 4 will not sit at T 4 or T 5. Find the number of ways they can occupy the vacant chairs.(8) 10). Using generating function, find the number of partitions of n=6. (8) 11). Find all partitions of n=7. (8) 12)Prove that, for any positive integer n,n!= 13) Determine the co-efficient of (a) w 3 x 2 yz 2 in (2w x + 3y 2z) 8. (b) a 2 b 2 c 2 d 5 in (a + 2b 3c + 2d + 5) 16. n k=0 ( n k )d k.

8 14) In how many ways can the integer 1,2,3.,10 be arranged in a line so that no even integer is in its natural position. 15). Determine the number of integer solutions of x 1+ x 2+ x 3+ x 4 =32, where (a) x i 0, 1 i 4 (b) x i > 0, 1 i 4 (c) x 1, x 2 5, x 3, x 4 7 (d) x i 8, 1 i 4 16).compute ϕ(n) for (i) n = 51, (ii) = ).Find the number of integer solutions of x 1+ x 2+ x 3+ x 4+ x 5 =30 Where x 1 2, x 2 3, x 3 4, x 4 2, x ). (i) How many arrangements are there of all the vowels adjacent in SOCIOLOGICAL? (4) (N/D-16) (ii) Find the value of n for the following: 2P(n,2)+50 = P(2n,2). (5) (N/D-16) (iii) How many distinct four digit intergers can one make from the digits 1,3,3,7,7 and 8? (4) (N/D-16) (iv) In how many possible ways could a student answer a 10-question true-false test? (3) (N/D-16) 19). (i) How many arrangements of the letters in MISSISSIPPI has no consecutive S s? (4) (N/D-16) (ii) A gym coach must select 11 seniors to play on a football team. If he can make his selection in 12,376 ways, how many seniors are eligible to play? (4) (N/D-16) (iii) How many permutations of size 3 can one produce with the letters m, r, a, f and t? (4) (N/D-16) (iv) Rama has two dozen each of n different colored beads. If she can select 20 beads (with repetitions of colors allowed), in 230,230 ways, what is the value of n? (4) (N/D-16) 20) In how many ways can the 26 letters of the alphabet be permuted so that the patterns car,dog, pun or byte occurs? Use the principle of inclusion and exclusion for this. (16) (A/M-17) 21). When n balls numbered 1, 2, 3 n are taken in succession from a container, a rencontre occurs if mth ball withdrawn is numbered m, 1 m n. Find the probability of getting (i) (ii) (iii) (iv) No rencontres. Exactly one rencontre Atleast one rencontre and r rencontries, 1 r n. Show intermediate steps. (16) (A/M-17)

9 Unit-V[Generating Functions] PART-A 1. Find the sequence generated by the following function (3 + x) Using exponential generating function, find the number of ways in which 5 of the letters in the word calculus be arranged. 3. Find the generating function of the sequence 0 2, 1 2, 2 2, 3 2,. 4. Solve the recurrencies a n+1 = 4a n for n 0 gvien that a 0 =3. 5. Find the sequence generated by the following function 1 1 x + 2x3. 6. Find the generating functions of the following sequence 1 2, 2 2, 3 2, 4 2,. 7. Determine the coefficient of x 27 in (x 4 + 2x 5 + 3x 6 + ) Determine the coefficient of x n in (x 8 + x 9 + x 10 + ) Find all partitions of Draw the Ferrers graph for the partition 20= Hence determine the conjugate of this partition. 11.Find the generating function of the following sequence 1,-1,1,-1,1,- 1, 12. Find the generating function of the following sequence 1,2,2 2, 2 3, 2 4,. 13. Find the generating function of the following sequence a, a 3, a 5, a 7,., a R. 14. Determine the sequence generated by the following exponential generating function f(x)=3e 3x. 15. Determine the sequence generated by the following exponential generating function f(x)= 1 1 x. 16. Find the general solution of the following recurrence relation a n+1= 3a n n 0 a 0= Find the general solution of the following recurrence relation a n= 7a n 1 n 1 a 2 = Solve: a n = na n 1 n 1 a 0 = Find : a 12 if a n+1 = 5a n where a n >0 for n 0 given that a 0 = Find the general term of the sequence 2, 10, 50, 250, 21. Find the general term of the sequence 6,-18, 54,-162, 22. Find the partition of Find the partition of Define recurrence relation. NOV/DEC Define generating function. NOV/DEC Give the explanation for the following: APR/MAY 2017

10 Generating function for the no. of ways to have n cents in pennies and nickels.=(1+x+x 2 + )(1+x 5 +x 10 + ) 27. Solve the recurrence relation an+1-an=3n 2 -n n 0 a0=3. APR/MAY 2017 PART-B 1.Solve the recurrence relation a n + a n 1 6a n 2 = 0, n 2, given that a 0 = 1 and a 1 = Solve the recurrence relation a n = 5a n 1 + 6a n 2 = 0, n 2, given that a 0 = 1 and a 1 = Solve the recurrence relation 2a n+2 11a n+1 + 5a n = 0, n 0, given that a 0 = 2 and a 1 = 8. 4.Solve the recurrence relation a n 6a n 1 + 9a n 2 = 0, n 2, given that a 0 = 5 and a 1 = Solve the recurrence relation a n = 2(a n 1 a n 2 ) = 0, n 2, given that a 0 = 1 and a 1 = 2. 6.If a 0 = 0,a 1 = 1, a 2 = 4,a 3 = 37 satisfy the recurrence relation a n+2 + ba n+1 + ca n = 0, where n 0 and b, c are constant,determine b,c and solve for a n. 7. Solve the recurrence relation 2a n+3 = a n+2 + 2a n+1 a n, n 0, given that a 0 = 0, a 1 = 1, a 2 = Solve the recurrence relation a n+2 5a n+1 + 4a 2 n = 0, n 0, given that a 0 = 4,a 1 = Solve the recurrence relation a n 3a n 1 = 5x3 n for n 1, given that a 0 = Using generating function, find the number of partitions of n=6. 11.Find all partitions of n= Solve the recurrence relation a n 3a n 1 = 5(7 n ), n 1, a 0 = Solve the recurrence relation F n+2 = F n+1 + F n = 0, n 0, given that F 0 = 0 and F 1 = Discuss about exponential generating function with an example. (10) (N/D-16) 15. Find the unique solution of the recurrence relation 6a n 7a n 1 = 0, n 1, a 3 = 343. (6) (N/D-16) 16. The population of Mumbai city is 6,000,000 at the end of the year The number of immigrants is20000 n at the end of year n. The population of the city increases at the rate of 5% per year. Use a recurrence relation to determine the population of the city at the end of (8) (N/D-16) 17. Write short notes on summation operator. (8) (N/D-16) 18. If an is count of number of ways a sequence of 1s and 2s will sum to n, for n θ. Eg a3=3 (i) 1, 1, 1; (ii) 1, 2 and (iii) 2, 1 sum up to 3. Find and solve a sequence relation for an. APR/MAY What are Ferrers diagrams? Describe how they are used to (i) represent integer partition (ii) Conjugate diagram or dual partitions (iii) self-conjugates (iv) representing bisections of two partition. APR/MAY 2017

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