Graph Theory & Combinatorics June 2012

7/29/2019 Graph Theory & Combinatorics June 2012

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USN I I I I I I I I I I IFourth Semester B.E. Degree Examination, June 2012

Graph Theory and Comb inatorics

06CS42

Tim e : 3 hrs . Max. Ma rk s :lOONote: An swer FIVE full questions, selectingat least TW O questions from each part.

PART- A1 a. Show that the max number of edges in a s imple gra ph w ith n ve rtices is n(n - I )/2 (06 Marks)

b. How ma ny ve1tices will the fo llow ing graphs have if they contain :i) 16 edges and a ll ve1tices of deg ree 4 .ii) 12 e dges, 6 ve rtices of deg ree 3. and o the r ve1ti ces of degree less than 3. (08 Marks)

c. Prove that numbe r of vert ices of odd deg ree in a graph is even. (06 Marks)2 a . Dete rmine the number of ve 1tices . numbe r of edges and the numbe r of reg ions fo r each of

the p lanar graph in the Fig.Q.2(a )( i) and Fig.Q.2(a)( ii) and also verify that Eulers theo re mfor connected plana r graph is satisfied. (06 Marks)

Fig.Q.2(a)( i) F ig.Q 2(a)( i i)b . F ind a Hamilton cycle, if ex ists, for each of the graph s shown in Fig.Q.2(b )( i) andF ig.Q.2(b)( ii ). If graph has no Hamil ton cyc le, determ ine whethe r it has a Hamilton path .(06 M a rks)

0... b c.

~ . .. Ja.

h \> t 0 tFig.Q.2( b)( i) Fig.Q.2(b )( ii)c. Dete1mine the chromat ic po lynomi als G A.) and chromatic numbe rs of each of the

follow in g g raphs Fig .Q.2(c)( i) and Fi g.Q.2(c)( ii ).

l.V ') t "'I ") .Fig.Q .2(c)(i) Fi g .Q.2(c) ( ii )

3 a. S ho w that for eve ry tree T = V, E) if lVI 2 , then T has at leas t two pendent ve rtices .(04 Ma rks)

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7/29/2019 Graph Theory & Combinatorics June 2012

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06CS42b. Suppose that a tree T has two vertices of degree 2, four vert ices of degree 3 and threevertices of degree 4. Find the number of pendent ve1t ices in T. (06 Marks )c. What is a prefix code? Construct an optimal prefix code for the symbols a, b, c, . . .. i, j thatoccur in a given sample with re spect to frequencies 78, 16, 30, 35, 125, 3 1, 20. 50, 80, 03.

(10 Marks )4 a. Explain Prim 's algorithm for finding shortest spanning tree of a weighted graph. (08 Marks)b. Defin e the following with respect to bipartite graph G = (V, E) :i) Comple te matching ; ii) Maximal matcing

ii ) Defici ency of graph G. (06 Marks)c. Define : i) Cutse t ; ii ) Edge -connectivity; iii ) Ye1tex - connectiv ity with one example foreach. (06 Marks)PART-B5 a. State two basic count ing princ iples . Give on example for each. (06 Marks)

b. A committee of 4 is to be chosen out of 6 Englishma n, 5 Frenchmen and 4 Ind ian s, thecommittee is to contain one of each nationalityi) In how many ways can it be done?ii ) In how many arrangements will a pa1ticular Indian be? (08 Marks)c. Cons ider the following program segment, whe re i,j and k are integer variablesfor i : = I to 20 dofor j : = I to i dofor k : = I to j doprint(i * j + k).Dete1mine how many times the print( ) statement is executed in above program. (06 Marks)

6 a. In how many ways can these integers I, 2, 3, . . .., 10 be arranged in a line so that no even

7

integer is in its natural pos ition. (06 Marks)b. Determine the number of positive integers n where I ::; n ::; I00 and n is not di visible by

2, 3, or 5. (08 Marks)c. A board consists of the shaded part as in the Fig.Q.6(c ). Find its Rook polynomial. (06 Marks)W/}1l!jj

Fig.Q.6(c) (/II! I;/;r{Za. Determine the coefficient of x8 in (x - 3)(x-2 / (08 Marks)b. Find the generating function for each of the following sequence :

i) 0, 2, 6, 12, 20, 30, 42, .. . .. .ii) 8, 26, 54, 92, ..... .c. Find a formu la to express 02 +

operator.(06 Marks)12 + 2 2 + ..... + n2 as a function of n, using summation(06 Marks)

8 a. Solve the relation Fn+2= Fn+l + F11 , where n 0 and Fo= 0, F 1= I. (06 Marks)b. Us ing generating function, solve Yn+2 - 4Yn+l + 3yn = 0 given that yo= 2, y, = 4. (08 Marks)c . Findthegenerated solution of S(K) - 3S(K - I ) -4S(K-2) = 4K where K ~ 2(06 Marks)

* * * * *


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Documents

Graph Theory & Combinatorics June 2012