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060010212-Advanced Mathematics for Computer Application 2016

Mr.NikhilChoksi Page1

Unit-1:MatrixAlgebraShortquestionsanswer

1. WhatisMatrix?2. Definethefollowingterms:

a) Elementsmatrixb) Rowmatrixc) Columnmatrixd) Diagonalmatrixe) Scalarmatrixf) UnitmatrixORIdentitymatrixg) Triangularmatrixh) Comparablematricesi) Equalityofmatricesj) Skew-symmetricmatrix

3. IfAisamatrixoforderp×qandBisamatrixoforderq×r,thenwhatistheorderoftheproductofmatrixAB?

4. Whatisthenecessaryconditionfortheadditionoftwomatrices?5. “Everyidentitymatrixisadiagonalmatrix”TrueorFalse?Justifyyouranswer.6. “Matrixmultiplicationisalwayscommutative”TrueorFalse?Justifyyouranswer.7. Whatisnecessaryconditionformatrixmultiplication?8. Findtheinverseofmatrixshownbelow.

2 00 0

9. Statethedifferencebetweenamatrixandadeterminant.10. Listoutspecialtypesofmatrices.11. Defineinverseofamatrix.12. Givetheconditionfortheexistenceoftheinverse.13. Listoutthepropertiesoftheinverseofamatrix.14. Whicharethepropertiesofthetransposeofamatrix?15. Whatdoyoumeanbytheprincipaldiagonalofthematrix?16. Whatistransposeofamatrix?

17. IfA= −3 83 5 andB= 3 −8

3 −5 ,findB-A.

18. IfP= 2 −3−1 9 ,findBT.

19. IfB= 1 04 7 ,find2Band-3B.

20. IfA= 2 31 4 andB= 5 1

0 3 ,findABandBA.IsAB=BA?

Longquestionsanswer

1. IfA=7 3 −50 4 21 5 4

andB=3A;C=B+2A-5I.FindmatrixDsuchthatD=2A+B-C.



2. IfA=6 3−3 912 −6

findthematrixBsuchthat2AT+3B=0.

3. FindABandBA.

WhereA=1 −1 1−3 2 −1−2 1 0

andB=1 2 32 4 61 2 3

4. IfA= 2 67 2 B= −3 5

0 8 andC= 4 79 5 provethatA(BC)=(AB)C.

5. IfA=1 30 2−1 4

andB= 1 2 3 −42 0 −2 1 findAB.

6. IfA=0 4 31 −3 −3−1 4 4

provethatA2=I.

7. IfA=1 2 22 1 22 2 1

satisfiesthefollowingequation:A2-4A-5I=O.WhereOisanullmatrix

andIisaunitmatrix.

8. IfA= 1 2 3 andB=456findABandBB’.

9. IfA= 1 −12 −1 ,B=

1 𝑥4 𝑦 and(A+B)2=A2+B2,findxandy.

10. Findadjointofthefollowingmatrices:

a) 2 −57 9

b) −4 38 7

c) 2 −1 31 1 11 −1 1

d) 5 −6 47 4 −32 1 6

11. Findtheinverseofthefollowingmatrices:

a) 2 35 −7

b) 2 −34 −1

c) 𝑎 𝑏𝑐 𝑑

d) −1 −2 3−2 1 14 −5 2



e) 2 −1 4−3 0 1−1 1 2

f) 1 1 11 2 −32 −1 3

12. IfA= 3 72 5 ,findA+AT+A-1.

13. IfA= 3 12 4 ,B= 5 6 0

0 1 2 showthat(AB)-1=B-1A-1.

14. IfA= 4 17 2 andAB= 1 0

0 1 ,findmatrixB.

15. Findthetransposeof:5 6 −7 5 04 3 0 1 2−6 2 1 −3 −4

16. IfA=2 5 72 −1 03 4 8

B=1 4 93 −2 4−5 6 8

verifythat(i)(A+B)T=AT+BTand(ii)(AB)T=BTAT.

17. FindtheinverseofthefollowingmatricesandverifythatAA-1=I.

a) 2 1 −11 0 −11 1 2

b) 6 34 5

18. Givethreedifferencesbetweendeterminantandmatrixeachwithexample.19. DiscusslawsofMatrixAlgebra.20. Explainspecialtypesofmatrices.

MultipleChoiceQuestions

1. IfA= 2 4 68 5 3 ,theorderofmatrixAis

a) 3×2b) 2×3c) 1×3d) 3×1

2. IfA= 1 2 3 ,whichtypeofthegivenmatrixB?a) Unitmatrixb) Rowmatrixc) Columnmatrixd) Squarematrix

3. MentionthetypeofthematrixA=1 −2 4−2 3 54 5 8

a) Symmetricmatrixb) Skew-symmetricmatrixc) Nullmatrixd) Identitymatrix



4. 0 00 0 ismatrixofthetype:

a) zeromatrixb) rowmatrixc) columnmatrixd) unitmatrix

5. 1 00 1 ismatrixofthetype:

a) zeromatrixb) unitmatrixc) rowmatrixd) columnmatrix

6. IfA=4 −5 20 6 92 7 8

,thediagonalelementsare:

a) 4,6,8b) 4,0,2c) 2,6,2d) Alloftheabove

7. IfA= 1 42 5 ,B=

−2 −13 0 then,A–2B–Igives

a) 4 64 6

b) −4 −6−4 −6

c) 4 6−4 6

d) 4 64 −6

8. IfB=2 4 14 −6 30 6 4

a) -2b) 2c) 6d) -6

9. IfA= 3 24 1 ,B= 1 0

3 1 ,theproductBAis

a) Noneoftheabove

b) 3 213 −7

c) 3 −213 −7

d) −3 213 7

10. IfA= 1 23 −4 ,thenadj.Ais:

a) −4 −2−3 −1



b) 4 23 1

c) −4 −2−3 1

d) 4 23 −1

11. IfB= 2 −31 6 ,thentransposeofBis:

a) 2 13 6

b) 2 1−3 6

c) 2 −31 6

d) 2 31 −6

12. IfA= 4 5−2 3 ,then(AT)Tis:

a) 4 −25 3

b) 4 5−2 3

c) 4 52 3

d) Noneoftheabove

13. IfA= 2 35 −2 ,thenA-1is:

a) (1/-19)Ab) Ac) (1/19)Ad) –A

14. IfA=1 −1 12 −1 01 0 0

,thenA3is:

a) I2b) I4c) I3d) Noneoftheabove

15. IfA= 𝑎 𝑏𝑐 𝑑 then,adj(adj(A))gives:

a) –Bb) adj.Bc) Bd) (1/2)B

16. IfA= 3 −62 −4 ,then|A|=

a) -12b) 12c) 0d) Noneoftheabove

Fillintheblanks1. A(n)__________________isanarrangementofnumbersinrowsandcolumns.



2. The individual quantities like a11, a22, a13.....amn are called the ___________ of thematrix.

3. Iftherearemrowsandncolumnsinthematrixthenorderofmatrixis___________.4. IfweinterchangerowsandcolumnsofamatrixA,thenewmatrixobtainedisknownas

the_______________ofmatrixAanditisdenotedby_____.5. A matrix in which there is only one row and any number of columns is said to be

__________matrix.6. A matrix in which there is only one column and any number of rows is said to be

___________matrix.7. Ifalltheelementsofmatrixarezeroisknownas_______matrix.8. Amatrixinwhichnumberofrowsandnumberofcolumnsare_________issaidtobea

squarematrix.9. If the transpose of a square matrix gives the same matrix is known as

________________matrix.10. Eachdiagonalelementsofaskewsymmetricmatrixmustbe__________.11. Asquarematrixinwhichalldiagonalelementsare___________andallotherelements

are___________isknownasanidentitymatrix.12. Ifallelementsexceptdiagonalelementsofasquarematrixarezerothematrixissaidto

be__________________matrix.13. Theadditionorsubtractionoftwoormorematrices ispossibleonlywhentheyareof

the_______________.14. IfAisamatrixoforderm×nandBisamatrixofordern×p,thenproductABwillbea

matrixoforder__________________.15. In_______________thenumberofrowsandcolumnsareequal.16. In_______________thenumberofrowsandcolumnsarenotnecessarilyequal.17. Adeterminanthas__________.18. Amatrixcannothave__________.19. The__________ofanyelementofamatrixcanbeobtainedbyeliminatingtherowand

columninwhichthatelementlies.20. __________ofasquarematrixisthetransposeofthematrixoftheco-factorsofagiven

matrix.True-False

1. IfAbeamatrixoforderof3×2,thenthereare3columnsand2rowsinthematrixA.2. Amatrixisanarrangementofnumbersinrowsandcolumns.3. Thematrixisdenotedbysmallletterslikea,b,cetc.4. Theelementofithrowandjthcolumnisdenotedbyaij.5. Twomatricesaresaidtobeequalonlyifthenumberofrowsinbothmatricesshouldbe

equal.6. Amatrixinwhichthereisonlyonecolumnandanynumberofrowsissaidtobearow

matrix.7. Amatrix in which there is only one column and any number of rows is said to be a

columnmatrix.



8. Azeromatrixcanbearowmatrixoracolumnmatrix.9. Eachdiagonalelementofaskewsymmetricmatrixmustbezero.10. Inaskewsymmetricmatrixdiagonalelementsarenon-zero.11. Ifallelementsexceptdiagonalelementsofasquarematrixarezerothematrixissaidto

beadiagonalmatrix.12. Theadditionorsubtractionoftwoormorematrices ispossibleonlywhentheyareof

thesameorder.13. Thescalarproductofamatrixisobtainedbymultiplyingeachelementoffirstrowofthe

matrixbythatscalar.14. ForthemultiplicationoftwomatricesAandB,thenumberofcolumnsofmatrixAand

thenumberofrowsofmatrixBshouldnotbeequal.15. Thematrixmultiplicationiscommutative.16. IfAandBanytwomatricesofanyorderm×n,thenA+B=B+A.17. IfA,B,cbeanythreematricesofthesameorderm×n,thenA+C=B+CgivesA=B.18. Adjoint of a squarematrix is the transpose of thematrix of the co-factors of a given

matrix.19. InEqualmatrices,theorderofbothmatricesisnotnecessarilyinthesameorder.20. Indeterminantthenumberofrowsandcolumnsareequal.21. Inmatrixthenumberofrowsandcolumnsarenotnecessarilyequal.22. Amatrixhasavalueandadeterminantcannothaveavalue.

Unit-2:GroupTheoryShortquestionsanswer

1. Whatisanalgebraicsystem?2. Definen-aryoperation.3. Statetheclosurepropertyofabinaryoperationwithanexample.4. Define identity, inverse and idempotent elements of an algebraic systemwith proper

examples.5. Whatishomomorphismwithrespecttoanalgebraicsystem?6. Defineisomorphismwithrespecttoanalgebraicsystem.7. WhatisBinaryoperation?8. Definesub-algebraicsystemwithanexample.9. Definedirectproductoftwoalgebraicsystems.10. Definesemigroupandmonoidwithanexampleforeach.11. Giveanexampleofasetwhichisclosedundercertainbinaryoperationbutasubsetof

whichmaynotclosedunderthesameoperation.12. Definesub-semigroupandsub-monoidwithanexampleforeach.13. Defineagroupwithanexample.14. Statethebasicpropertiesofagroup.15. Definetheorderofagroupandorderofanelementofagroup.16. Defineadihedralgroup.



17. Givenameofsomepropertiessatisfiedbyalgebraicsystems.18. Defineacyclicwithanexample.19. Howmanygeneratorsarethereforacyclicgroupoforder8?Whatarethey?20. Defineapermutationgroup.

Longquestionsanswer1. If*isthebinaryoperationonthesetRofrealnumbersdefinedbya*b=a+b+2ab,

a) Findif{R,*}isasemigroup.Isitcommutative?b) Findtheidentityelement,ifexists.c) Whichelementshaveinversesandwhatarethey?

2. IfNisthesetofpositiveintegersand*denotestheleastcommonmultipleonN,showthat{N,*}isacommutativesemigroup.Findtheidentityelementof*.WhichelementsinNhaveinversesandwhatarethey?

3. IfQisthesetofrationalnumbersand*istheoperationonQdefinedbya*b=a+b–ab,showthat{Q,*}isacommutativesemigroup.Findalsotheidentityelementof*.FindtheinverseofanyelementofQifitexists.

4. If*istheoperationdefinedonS=Q×Q,thesetoforderedpairsofrationalnumbersandgivenby(a,b)*(x,y)=(ax,ay+b),

a) Findif(S,*)isasemigroup.Isitcommutative?b) FindtheidentityelementofS.c) Whichelements,ifany,haveinversesandwhatarethey?

5. IfS=N×N,thesetoforderedpairsofpositiveintegerswiththeoperation*definedby(a,b)*(c,d)=(ad+bc,bd)andiff:(S,*)→(Q,+)isdefinedbyf(a,b)=a/b,showthatfisasemigrouphomomorphism.

6. ShowthatthesetQ+ofallpositiverationalnumbersformsanabeliangroupundertheoperation*definedbya*b=1/2ab;a,b∈Q+.

7. Provethattheset{1,3,7,9}isanabeliangroupundermultiplicationmodulo10.8. Showthattheset{Zm}ofequivalenceclassesmodulomisanabeliangroupunderthe

operation+mofadditionmodulom.9. If {G,*} isanabeliangroup,showthat (a*b)n=an*bn, foralla,b∈ G,wheren isa

positiveinteger.10. Definethedihedralgroup(D4,*)andgiveitscompositiontable.11. Showthat,if{Un}isthesetofnthrootsofunity,{Un,X}isacyclicgroup.Isitabelian?12. Showthateverygroupoforder3iscyclicandeverygroupoforder4isabelian.13. Showthatthesetofallpolynomialsinxundertheoperationofadditionisagroup.14. Prove that the set {0, 1, 2, 3, 4} is a finite abelian group of order 5 under addition

modulo5ascomposition.15. If*isdefinedonQ+suchthata*b=ab/3,fora,b∈Q+,showthat{Q+,*}isanabelian

group.16. Showthatthegroup{(1,2,4,5,7,8),X9}iscyclic.Whatareitsgenerators?17. If*isdefinedonZsuchthata*b=a+b+1fora,b∈Z,showthat{Z,*}isanabelian

group.18. Showthatthegroup{(1,3,3,4,5,6),X7}iscyclic.Howmanygeneratorsaretherefor

thisgroup?Whatarethey?



19. IfR is the setof realnumbersand* is theoperationdefinedbya*b=a+b+3ab,where a, b ∈ R, show that {R, *} is a commutative monoid. Which elements haveinversesandwhatarethey?

20. IfZ6isthesetofequivalenceclassesgeneratedbytheequivalencerelation“congruencemodulo6”,provethat{Z6,X6}isamonoidwheretheoperationX6andZ6isdefinedas[i] X6 [j] = [(i X j) (mod 6)], for any [i], [j] ∈ Z6. Which elements of the monoid areinvertible?


1. UsualSubtractionisnotbinaryoperationonthesetisa) Zb) Rc) Nd) Q

2. (R,X)isnotGroupbecausea) XisnotassociativeonRb) identityelementdoesnotexistsinRwithrespecttoXc) inversionpropertyisnotsatisfied

3. Ifagroupsatisfiedcommutativepropertythenitisknownasa) abeliangroupb) symmetricgroupc) semigroupd) monoid

4. Asemigroup(G,*)issaidtobemonoidifa) *isassociativeb) *iscommutativec) Thereexistsidentityelementwithrespectto*.d) EveryelementofGhasinversewithrespectto*.

5. Anisomorphismg:{X,•}→{Y,*}iscalledanautomorphismif:a) Y=Xb) Y→Xc) X↔Yd) X≤Y

Fillintheblanks

1. Asystemconsistingofanon-emptysetandoneormoren-aryoperationsonthesetiscalleda(n)_________________.

2. Analgebraicsystemwillbedenotedby_______________.3. IfSisanon-emptysetand*beabinaryoperationonS,thenthealgebraicsystem{S,*}

iscalled_____________,iftheoperation*is______________.4. If a semi group {M, *} has an identity elementwith respect to the operation *, then

{M,*}iscalled________________.5. Theidentityelementofagroup(G,*)is___________.



6. Theinverseofeachelementof(G,*)is____________.7. A(n)_______________mappingofanon-emptysetS→SiscalledapermutationofS.8. ThesetofGofallpermutationsonanon-emptysetSunderthebinaryoperation*of

therightcompositionofpermutationsisagroup{G,*}calledthe_____________group.9. Thedihedralgroupisdenotedby____________.10. The set of transformations due to all rigidmotions of a regularmotions of a regular

polygonofnsidesresultinginidenticalpolygonsbutwithdifferentvertexnamesunderthebinaryoperationofrightcomposition*isagroupcalled_________________.

11. Agroup{G,*}issaidtobe______________,ifthereexistsanelementa∈ GsuchthateveryelementxofGcanbeexpressedasx=anforsomeintegern.

12. The cyclic group is generated by a or a is a generator of G, which is denoted by_____________.

13. Acyclicgroupis___________.14. A group homomorphism f is called group isomorphism, if f is _____________ and

_____________.15. Anisomorphismg:{X,•}→{Y,*}iscalledanautomorphism,if______________.

True-False

1. Everymonoidissemigroup.2. Everybinaryoperationsatisfiedclosureproperty.3. Inagrouptheremaybeanelementwhichhastwoinverses.4. Ifthehomomorphismg:{X,•}→{Y,*}isone-one,thegiscalledanepimorphism.5. Ifthehomomorphismg:{X,•}→{Y,*}isonto,thengiscalledamonomorphism.6. Ifg:{X,•}→{Y,*}isone-oneandontothengiscalledanisomorphism.7. Ahomomorphismg:{X,•}→{Y,*}iscalledanendomorphism,ifY=X.8. Anisomorphismg:{X,•}→{Y,*}iscalledanautomorphism,ifY=X.9. Compositionoftwohomomorphismsisalsoahomomorphism.10. Thesetofallsemigroupendomorphismofasemigroupisnotasemigroupunderthe

operationofcomposition.11. Thesetofidempotentelementsofacommutativemonoid{M,*,e}formsasubmonoid

ofM.12. Theidentityelementofagroup(G,*)isnotunique.13. Theinverseofeachelementof(G,*)isunique.14. Thecancellationlawsarefalseinagroup.15. (a*b)-1=b-1*a-1.16. (G,*)canhaveanidempotentelementexcepttheidentityelement.17. Acyclicgroupisabelian.18. Ifaisageneratorofacyclicgroup{G,*},a-1isnotageneratorof{G,*}.19. Agrouphomomorphismfiscalledgroupisomorphism,iffisone-to-oneandonto.20. IfSisanon-emptysetand*beabinaryoperationonS,thenthealgebraicsystem{S,*}

iscalledsemigroup,iftheoperation*iscommutative.



Unit-3:BasicsofGraphTheoryShortquestionsanswer

1. Define:Simplegraph.2. Whatdoyoumeanbymultigraph?3. Whatispseudograph?Giveanexample.4. Whatdoyoumeanbydegreeofavertex?5. Whatarethedegreesofanisolatedvertexandapendantvertex?6. Statethehand-shakingtheorem.7. Definein-degreeandout-degreeofavertex?8. Whatismeantbysourceandsinkingraphtheory?9. Definecompletegraphandgiveanexample.10. Defineregulargraph.Canaregulargraphcanbeacompletegraph?11. Canacompletegraphbearegulargraph?Establishyouranswerby2examples.12. Definen-regulargraph.Giveoneexampleforeachof2-regularand3-regulargraphs.13. Defineabipartitewithanexample.14. Inwhatwayacompletelybipartitegraphdiffersfromabipartitegraph?15. DrawK5andK6.16. Defineasubgraphandspanningsubgraph.17. Whatisaninducedsubgraph?18. DrawK2,3andK3,3graphs.19. WhatdoyoumeanbypropersubgraphofG?20. Giveanexampleofpendentvertex.

Longquestionsanswer

1. Determinewhetherthefollowinggraphsshownhasdirectedorundirectededges,whetherithasmultipleedges,orwhetherithasoneormoreloops?

a) b)c)d)

2. Determinewhetherthefollowinggraphsshownhasdirectedorundirectededges,

whetherithasmultipleedges,orwhetherithasoneormoreloops?a) b)c)

3. Doesthereexist,asimplegraphwith5verticesofthegivendegreesbelow?Ifsodraw



suchagraph.(a) 1,2,3,4,5;(b)1,2,3,4,4;(c)3,4,3,4,3;(d)0,1,2,2,3;(e)1,1,1,1,1.

4. Findthenumberofvertices,numberofedges,anddegreeofeachvertexinthe

followingundirectedgraphs.Identifyallisolatedandpendantvertices.

5. Findthenumberofvertices,numberofedges,anddegreeofeachvertexinthe

followingundirectedgraphsandhence,verifytheHandshakingTheoremforeachone:



6. Findthenumberofvertices,numberofedges,in-degreeandout-degreeofeachvertexinthefollowingdirectedgraphsandhence,verifythatthesumofin-degrees,thesumofoutdegreesoftheverticesandthenumberofedgesinthefollowinggraphsareequal.

7. Drawthegraphs:K7,K1,8,K4,4,C7,W7,Q4.8. Howmanysub-graphswithat-leastonevertexdo(a)K2(b)K3,(c)W3have?9. Determinewhethereachofthesedegreesequencesrepresentsasimplegraphsornot?

Ifthegraphispossible,drawsuchagraph.(a) 5,4,3,2,1,0;(b)6,5,4,3,2,1;(c)2,2,2,2,2,2;(d)3,3,3,2,2,2;(e)3,3,2,2,2,2;

(f)1,1,1,1,1,1;(g)5,3,3,3,3,3;(h)5,5,4,3,2,1;(i)3,3,3,3,2;(j)4,4,3,2,1.10.Representthefollowinggraphsusingadjacentmatrices:

(a) (b)(c)(d)



(e) (f) (g)

(h) (i) (j) (k)

(l) (m) (n)


1. Agraph,inwhichthereisonlyanedgebetweenpairofvertices,iscalledaa) simplegraphb) pseudographc) multigraphd) weightedgraph

2. Whichsub-graphofgraphGneedsnotcontainallitsedges?a) Properb) Spanningc) Induced



d) Vertexdeleted3. TheHandshakingtheoremistrueforwhichofthefollowinggraphs.

a) Directedb) Undirectedc) Completed) Directedsimple

4. Theadjacencymatricesoftwographsareidenticalonlyifthegraphsare:a) simpleb) isomorphicc) bipartited) complete

5. Agraphinwhichloopsandparalleledgesareallowediscalled:a) simplegraphb) pseudographc) multigraphd) weightedgraph

6. Graphinwhichanumberisassignedtoeachedgearecalled:a) regulargraphb) pseudographc) multigraphd) weightedgraph

7. Ifthereispendentvertexingraph,ifthedegreeofavertexis:a) zerob) twoc) oned) four

8. Avertexwithzeroindegreeiscalled:a) sourceb) sinkc) pendentvertexd) isolatedvertex

9. Avertexwithzerooutdegreeiscalled:a) sourceb) sinkc) pendentvertexd) isolatedvertex

10. Asimplegraphinwhichthereisexactlyoneedgebetweeneachpairofdistinctvertices,iscalled:a) regulargraphb) simplegraphc) completegraphd) bipartitegraph



Fillintheblanks1. ______________arediscrete structures consistingof verticesandedges that connect

thesevertices.2. A node of a graph which is not adjacent to any other node is called a(n)

________________node.3. Agraphcontainingonlyisolatednodesiscalleda(n)_____________graph.4. IfingraphG=(V,E),eachedgee∈Eisassociatedwithanorderedpairofvertices,then

Giscalleda________________.5. If each edge is associated with an unordered pair of vertices, then G is called a(n)

________________graph.6. Anedgeofagraphthatjointsavertextoitselfiscalleda(n)_______________.7. If, inadirectedorundirectedgraph,certainpairsofverticesare joinedbymore than

oneedge,suchedgesarecalled_______________edges.8. A graph, in which there is only one edge between a pair of vertices, is called

an________________graph.9. Agraphwhichcontainssomeparalleledgesiscalledan________________.10. A graph in which loops and parallel edges are allowed is called an

___________________.11. Graphsinwhich______________isassignedtoeachedgearecalledweightedgraphs.12. Thedegreeofanisolatedvertexis______________.13. Ifthedegreeofavertexis______________,itiscalledapendantvertex.14. Thenumberof edgeswith v as their initial vertex is called the_____________and is

denotedas_________________.15. A vertex with zero in degree is called an ____________ and a vertex with zero out

degreeiscalledan________________.16. Asimplegraph,inwhichthereisexactlyoneedgebetweeneachpairofdistinctvertices

iscalleda_______________graph.17. Ifeveryvertexofasimplegraphhasthe__________degree,thenthegraphiscalleda

regulargraph.18. IfeachvertexofV1isconnectedwitheveryvertexofV2byanedge,thenGiscalledan

_________________________graph.19. In adirectedgraph, thenumberof edgeswith v as their terminal vertex is called the

_____________andisdenotedas______________.20. Thepairofnodesthatareconnectedbyanedgearecalled________________.

True-False

1. Asimplegraphhasnoloops;eachdiagonalentryoftheadjacencymatrixis0.2. Inloop,theinitialandterminalnodesareoneandthesame.3. A rowwith all 0 entries in an incidencematrix of a graph corresponds to a pendant

vertex.4. Acompletelybipartitegraphneednotbeasimplegraph.5. Allvertexdeletedsub-graphsarespanningsub-graphs.6. Theadjacencymatrixofapseudo-graphisasymmetricmatrix.



7. Thereare18sub-graphsofK3containingat-leastonevertex.8. Thepairofnodesthatareconnectedbyanedgearecalledadjacentnodes.9. Anodeofagraphwhichisnotadjacenttoanyothernodeiscalledanisolatednode.10. Agraphcontainingonlyisolatednodesiscalledanundirectedgraph.11. Ifeachedge isassociatedwithanunorderedpairofvertices, thengraphG iscalleda

directedgraph.12. Anedgeofagraphthatjointsanothervertexiscalledaloop.13. In a directed graph, certain pairs of vertices are joined bymore than one edge; such

edgesarecalledparalleledges.14. Agraphinwhichthereismorethanoneedgebetweenpairofverticesiscalledasimple

graph.15. Agraphwhichcontainssomeparalleledgesiscalledamultigraph.16. Agraphinwhichloopsandparalleledgesareallowediscalledaweightedgraph.17. Graphsinwhichanumberisassignedtoeachedgearecalledpseudograph.18. Ifthedegreeofavertexisoneormore,itiscalledapendantvertex.19. Avertexwithzeroindegreeiscalledasink.20. Avertexwithzerooutdegreeiscalledasource.21. Asimplegraph,inwhichthereisexactlytwoedgebetweeneachpairofdistinctvertices

iscalledacompletegraph.22. Ifeveryvertexofasimplegraphhasthesamedegree,thenthegraphiscalledaregular

graph.Unit-4:GraphsIsomorphism,PathandCircuitsShortquestionsanswer

1. Definegraphisomorphism.2. Giveanexampleofisomorphicgraphs.3. Whatistheinvariantpropertyofisomorphicgraphs?4. Give an example to show that the invariant conditions are not sufficient for graph

isomorphism.5. Defineapathandthelengthofapath.6. Whenisapathsaidtobesimplepath?Giveanexampleforeachofgeneralpathand

simplepath.7. Defineacircuit.Whenisitcalledasimplecircuit?8. Defineaconnectedgraphandadisconnectedgraphwithexample.9. Whatdoyoumeanbyconnectedcomponentsofagraph?10. Statetheconditionfortheexistenceofapathbetweentwoverticesinagraph.11. Findthemaximumnumberofedgesinasimpleconnectedgraphwithnvertices.12. Howwillyoufindthenumberofpathsbetweenanytwoverticesofagraphanalytically?13. DefineEulerianpathandEuleriancircuitofagraph,withanexampleforeach.14. StatethenecessaryandsufficientconditionfortheexistenceofanEuleriancircuitina

connectedgraph.15. WhenagraphiscalledanEuleriangraph?16. DefineHamiltonianpathandHamiltoniancircuitwithexamples.



17. WhenagraphiscalledHamiltoniangraph?18. Stateaninvariantintermsofcircuitsoftwoisomorphicgraphs.19. State thenecessaryand sufficient condition for theexistenceofanEulerianpath ina

connectedgraph.20. Givethedefinitionofastronglyconnecteddirectedgraphwithanexample.21. Defineweaklyconnecteddirectedgraphwithanexample.22. Whatismeantbyastronglyconnectedcomponentofadiagraph?23. What are the advantages ofWarshall’s algorithmoverDijkstra’s algorithm for finding

shortestpathsinweightedgraph?24. FindthenumberofHamiltoniancircuitsinK33.

Longquestionsanswer

1. Representeachofthesegraphswithadjacencymatrices:(a) K4;(b)K1,4;(c)K2,3;(d)C4;(e)W4;(f)Q3

2. Drawthegraphswiththegivenadjacencymatrix:(a)(b)(c)(d)(e)(f)

(g)

(h)(i)

3. Representtheincidencematrixforthefollowinggraphs:

(a) (b) (c) (d)



(e)(f)(g)

4.Determinewhetherthegivengraphsareisomorphicornot?JustifybygivingappropriateReasons(a)

(b)

(c)

(d)



(e)

(f)

(g)(h)

(i)

(j)(k)



11. Determinewhetherthegivenpairofdirectedgraphsisisomorphic?

(a) (b)

(c)(d)

12. Drawacallgraphforthetelephonenumbers555-0011,555-1221,555-1333,555-8888,555-2222,555-0091,and555-1200iftherewerethreecallsfrom555-0011to555-8888,twocallsfrom555-8888to555-0011andtwocallsfrom555-2222to555-0091twocallsfrom555-1221toeachoftheothernumbers,andonecallfrom555-1333toeachof555-0011,555-1221and555-1200

13. Drawthegraphsrepresentedbythefollowingincidencematrices:



14. GivethestepbystepprocedureofDijktra’salgorithmtofindtheshortestpathbetween

anytwovertices.15. Findtheshortestpathfromtheverticesatozinthefollowinggraphmodel:

16. Findtheshortestpathforthegraphmodelgivenbelow:

17. Findtheshortestpathfromtheverticesatozinthefollowinggraphmodel:

18. Findtheshortestpathfromtheverticesatozinthefollowinggraphmodel:

19. ThediagrambelowshowsroadsconnectingtownsneartoRochdale.Thenumberson



eacharcrepresentthetime,inminutes,requiredtotravelalongeachroad.PeterisdeliveringbooksfromhisbaseatRochdaletoStockport.UseDijkstra’salgorithmtofindtheminimumtimeforPeter’sjourney.

20. ThediagrambelowshowsroadsconnectingvillagesneartoRoyton.Thenumberson

eacharcrepresentthedistance,inmiles,alongeachroad.LeonlivesinRoytonandworksinAshton.UseDijkstra’salgorithmtofindtheminimumdistanceforLeon’sjourneytowork.


1. Amatrixwhoserowsaretherowsoftheunitmatrix,butnotnecessarilyintheirnaturalorder,iscalled:

a) adjacencymatrixb) adjacencymatrixofpseudographc) adjacencymatrixofsimplegraphd) permutationmatrix

2. Iftheedgesinapatharedistinct,itiscalled:a) simplepathb) simplegraphc) lengthofthepathd) simplecycle

3. ThemaximumnumberofedgesinasimpledisconnectedgraphGwithnverticesandkcomponentsis:

a) (n-k)/2b) (n-k+1)/2



c) (n-k)(n-k+1)/2d) (n-k)(n-k-1)/2

4. Whichofthefollowingstatementistrueforconnectedgraph:a) AconnectedgraphcontainsanEulerpath,ifandonlyifeachofitsverticesisof

evendegree.b) AconnectedgraphcontainsanEulercircuit,ifandonlyifeachofitsverticesisof

odddegree.c) A connected graph contains an Euler path, if and only if it has exactly four

verticesofodddegree.d) A connected graph contains an Euler path, if and only if it has exactly two

verticesofodddegree.5. Adirectedgraphissaidtobestronglyconnectedif:

a) Thereisapathbetweeneverytwoverticesintheunderlyingundirectedgraph.b) Foranypairof verticesof thegraph,at leastoneof theverticesof thepair is

reachablefromtheothervertex.c) Theremustbeasequenceofdirectededgesfromanyvertexinthegraphtoany

othervertex.d) There is alwaysapathbetweenevery twoverticeswhen thedirectionsof the

edgesaredisregarded.

Fillintheblanks

1. TwographsG1andG2aresaidtobe________________toeachother,ifthereexistsa0ne-to-one correspondence between the vertexes sets which preserves adjacency ofthevertices.

2. Twographsareisomorphic,ifandonlyiftheirverticescanbelabelledinsuchawaythatthecorrespondingadjacencymatricesare___________.

3. An ____________in a graph is a finite alternating sequence of vertices and edges,beginning and ending with vertices, such that each edge is incident on the verticesprecedingandfollowingit.

4. Iftheedgesinapatharedistinct,itiscalleda_____________path.5. Thenumberofedgesinapathiscalledthe______________ofthepath.6. Ifthe______________and______________verticesofapatharethesame,thepathis

calledacircuitorcycle.7. Agraphthatisnotconnectediscalled________________.8. Thedisjointconnectedsubgraphsarecalledthe_____________ofthegraph.9. ApathofgraphGiscalleda_______________path,ifitincludeseachedgeofGexactly

once.10. A circuit of graph G is called a ______________ circuit, if it includes each edge of G

exactlyonce.11. A connected graph contains an Euler circuit, if and only if each of its vertices is of

____________degree.12. AconnectedgraphcontainsanEulerpath, if andonly if ithasexactly twoverticesof



___________degree.13. ApathofagraphG iscalleda________________path, if it includeseachvertexofG

exactlyonce.14. AcircuitofagraphGiscalleda________________circuit,ifitincludeseachvertexofG

exactlyonce,exceptthestartingandendverticeswhichappeartwice.15. AgraphcontainingaHamiltoniancircuitiscalleda________________graph.16. Agraphmaycontain__________________Hamiltoniancircuit.17. A directed graph is said to be _____________, if there is a path between every two

verticesintheunderlyingundirectedgraph.18. Asimpledirectedgraphissaidtobe________________,ifforanypairofverticesofthe

graph,atleastoneoftheverticesofthepairisreachablefromtheothervertex.19. ThemaximumnumberofedgesinasimpledisconnectedgraphGwithnverticesandk

componentsis______________________.20. Anystronglyconnecteddirectedgraphisalso______________.21. A_______________pathbetweentwovertices inaweightedgraph isapathof least

weight.22. Ina _________________graph,ashortestpathmeansonewith the leastnumberof

edges.True-False

1. Twographswithsameverticesandsameedgesarealwaysisomorphic.2. Thenumberofverticesofodddegreeinanundirectedgraphiseven.3. InthegraphK3,3thedegreeofeachvertexinthegraphisdegree3.4. Twographsareisomorphic,ifandonlyiftheirverticescanbelabelledinsuchawaythat

thecorrespondingadjacencymatricesaredifferent.5. Amatrix,whoserowsaretherowsoftheunitmatrix,butnotnecessarilyintheirnatural

order,iscalledapermutationmatrix.6. Iftheedgesinapathareequal,itiscalledasimplepath.7. Thenumberofverticesinapathiscalledthelengthofthepath.8. Iftheinitialandfinalverticesofapatharethedistinct,thepathiscalledacycle.9. Asimpledirectedgraphissaidtobestronglyconnected,ifforanypairofverticesofthe

graph,atleastoneoftheverticesofthepairisreachablefromtheothervertex.10. If the initial and final vertices of a simple path of non-zero length are the same, the

simplepathiscalledasimplecircuit.11. Agraphthatisnotconnectediscalleddisconnected.12. Adisconnectedgraphistheintersectionoftwoormoreconnectedsubgraphs.13. ThemaximumnumberofedgesinasimpledisconnectedgraphGwithnverticesandl

componentsis(n-k)/2.14. ApathofgraphGiscalledanEulerianpath,ifitincludeseachedgeofGexactlyonce.15. AconnectedgraphcontainsanEulercircuit, ifandonly ifeachof itsvertices isofodd

degree.16. AconnectedgraphcontainsanEulerpath, if andonly if ithasexactly twoverticesof

evendegree.



17. AgraphcontainsexactlyoneHamiltoniancircuit.18. Adirectedgraph issaidtobeweaklyconnected, if there isapathbetweeneverytwo

verticesintheunderlyingundirectedgraph.19. AHamiltonianpathcontainsaHamiltoniancircuit,butagraphcontainingaHamiltonian

pathneednothaveaHamiltoniancircuit.20. Ashortestpathbetweentwoverticesinaweightedgraphisapathofleastweight.21. Inanunweightedgraph,ashortestpathmeansonewiththemostnumberofedges.

Unit-5:TreesShortquestionanswer

1) Definefollowing.a. Treeb. Forestc. RootedTreed. Sub-treee. Parentofvertexvinarootedtreef. Childofvertexvinarootedtreeg. Siblingofvertexvinarootedtreeh. Ancestorofvertexvinarootedtreei. Descendantofvertexvinarootedtreej. Internalvertexk. Leafl. Levelofavertexm. Heightofavertexn. m-arytreeo. Fullm-arytreep. Completem-arytree

2) Giveanexampleofafull3-arytree.3) Howmanyleavesandinternalverticesdoesafullbinarytreewith25totalvertices

have?4) Whatarethemaximumandminimumheightsofabinarytreewith25verticeshave?5) Ineachofthefollowingcases,statewhetherornotsuchatreeispossible.

i. Abinarytreewith35leavesandheight100.ii. Afullbinarytreewith21leavesandheight21.iii. Abinarytreewith33leavesandheight5.iv. Arootedtreeofheight5whereeveryinternalvertexhas3childrenand

thereare365vertices.6) Giventwoverticesinatree,howmanydistinctsimplepathscanwefindbetweenthe

twovertices?7) Statefewpropertiesoftree.8) Howmanyverticesarethereinatreewith19edges?



9) Howmanyedgesarethereinatreewith16vertices?10) Howmanyverticesarethereinatreewith20edges?11) DoesthereexistatreeTwith8verticessuchthatthetotaldegreeofGis18?Justify

youranswer.12) Giveanexampleofagraphthatsatisfiesthespecifiedconditionorshowthatnosuch

graphexists.13) Atreewithsixverticesandsixedges.14) Atreewiththreeormorevertices,twoverticesofdegreeoneandalltheothervertices

withdegreethreeormore.15) Adisconnectedgraphwith10verticesand8edges.16) Adisconnectedgraphwith12verticesand11edgesandnocycle.17) Atreewith6verticesandthesumofthedegreesofallverticesis12.18) Aconnectedgraphwith6edges,4vertices,andexactly2cycles.19) Agraphwith6vertices,6edgesandnocycles.

LongQuestionanswer1) Whichofthegivengraphsaretrees?

2) Whichofthegivengraphsaretrees?

3) Answerthefollowingquestionsabouttherootedtreesgivenbelow:

(a)Whichvertexistheroot? (b)Whichverticesareinternal?



(c)Whichverticesareleaves? (d)Whichverticesarechildrenofj?4) Constructacompletebinarytreeofheight4andacomplete3-arytreeofheight3.5) Answerthefollowingquestionsabouttherootedtreesgivenbelow:

(a)Whichverticesaredescendantsofb?(b)Istherootedtreeafullm-arytreeforsomepositiveintegerm?(c)Whatisthelevelofeachvertexoftherootedtree?

(d)Drawasub-treeofthetreeinthegiventreefiguresthatisrootedat(a)a;(b)c6) Howmanyedgesdoesatreewith10,000verticeshave?7) Howmanyverticesdoesafullybinarytreewith1000internalverticeshave?8) Howmanyleavesdoesafully3-arytreewith100verticeshave?9) Howmanyedgesdoesafullbinarytreewith1000internalverticeshave?10) Drawthesubtreeofthetreethatisrootedat“a,c,e”

11) Drawthesubtreeofthetreethatisrootedat“a,c,e”

12) Whatisthelevelofeachvertexoftherootedtree



13) Whatisthelevelofeachvertexoftherootedtree

14) Supposethereexistsasimpleconnectedgraphwith16verticesthathas15edges.Does

itcontainavertexofdegree1?Justifyyouranswer.15) Drawatreewithtwoverticesofdegree3.Findthenumberofverticesofdegree1in

yourtree.16) Drawagraphhavingthegivenpropertiesorexplainwhynosuchgraphexists.

a) Tree;sixverticeshavingdegree1,1,1,1,3,3b) Tree;allverticesofdegree2c) Tree;10verticesand10edgesd) Connectedgraph;7vertices,7edgese) Tree;5vertices,totaldegreeoftheverticesis10

Multiplechoicequestion1.Atreeis

A)Anygraphthatisconnectedandeveryedgeisabridge.B)Anygraphthathasnocircuits.C)Anygraphwithonecomponent.D)Anygraphthathasnobridges.

2.Graph1isconnectedandhasnocircuits.Graph2issuchthatforanypairofverticesinthegraphthereisoneandonlyonepathjoiningthem.

A)Graph1cannotbeatree;Graph2cannotbeatree.B)Graph1mustbeatree;Graph2mayormaynotbeatree.C)Graph1mustbeatree;Graph2mustbeatree.D)Graph1mustbeatree;Graph2cannotbeatree.

3.SupposeTisatreewith21vertices.ThenA)Thasonebridge.B)Thasnobridges.C)Tcanhaveanynumberofbridges.D)Thas20bridges.

4.Inatreebetweeneverypairofverticesthereis?A)Exactlyonepath



B)Aself-loopC)TwocircuitsD)nnumberofpaths

5.Ifatreehas8verticesthenithasA)6edgesB)7edgesC)9edgesD)Noneoftheabove

6.AgraphistreeifandonlyifA)IsplanarB)ContainsacircuitC)Isminimally

D)Iscompletelyconnected7.SupposethatAisanarraystoringnidenticalintegervalues.WhichofthefollowingsortingalgorithmshasthesmallestrunningtimewhengivenasinputthisarrayA?

A)InsertionSortB)SelectionSortC)Ordered-dictionarysortimplementedusinganAVLtree.D)Ordered-dictionarysortimplementedusinga(2,4)-tree.

8.Howmanydifferent(2,4)-treescontainingthekeys1,2,3,4,and5exist(eachkeymustappearsonceineachoneofthese(2,4)-trees)?

(A)2 (B)3 (C)4 (D)59.ForwhichofthefollowingdoesthereexistagraphG=(V,E,φ)satisfyingthespecified

conditions?(A)Atreewith9verticesandthesumofthedegreesofalltheverticesis18.(B)Agraphwith5components12verticesand7edges.(C)Agraphwith5components30verticesand24edges.(D)Agraphwith9vertices,9edges,andnocycles.

10.ForwhichofthefollowingdoesthereexistasimplegraphG=(V,E)satisfyingthespecifiedconditions?

(A)Ithas3components20verticesand16edges.(B)Ithas6vertices,11edges,andmorethanonecomponent.(C)Itisconnectedandhas10edges5verticesandfewerthan6cycles.(D)Ithas7vertices,10edges,andmorethantwocomponents.

11.Forwhichofthefollowingdoesthereexistatreesatisfyingthespecifiedconstraints?(A)Abinarytreewith65leavesandheight6.(B)Abinarytreewith33leavesandheight5.(C)Afullbinarytreewithheight5and64totalvertices.(D)Arootedtreeofheight3,everyvertexhasatmost3children.Thereare40total

vertices. 12.Forwhichofthefollowingdoesthereexistatreesatisfyingthespecifiedconstraints?

(A)Afullbinarytreewith31leaves,eachleafofheight5.(B)Arootedtreeofheight3whereeveryvertexhasatmost3childrenandthereare41

totalvertices.(C)Afullbinarytreewith11verticesandheight6.(D)Abinarytreewith2leavesandheight100.

13.Thenumberofsimpledigraphswith|V|=3is



(a)29(b)28(c)27(d)26

14.Thenumberofsimpledigraphswith|V|=3andexactly3edgesis(a)92(b)88(c)80(d)84

15.Thenumberoforientedsimplegraphswith|V|=3is(a)27(b)24(c)21(d)18

16.Thenumberoforientedsimplegraphswith|V|=4and2edgesis(a)40(b)50(c)60(d)70

17.Whichdatastructureallowsdeletingdataelementsfromfrontandinsertingatrear?(a)Stacks(b)Queues(c)Deques(d)Binarysearchtree

18.Torepresenthierarchicalrelationshipbetweenelements,whichdatastructureissuitable?(a)Deque (b)Priority (c)Tree (d)Allofabove

19.Abinarytreewhoseeverynodehaseitherzeroortwochildreniscalled(a)Completebinarytree(b)Binarysearchtree(c)Extendedbinarytree(d)Noneofabove

20.Abinarytreecaneasilybeconvertedintoq2-tree(a)byreplacingeachemptysubtreebyanewinternalnode(b)byinsertinganinternalnodesfornon-emptynode(c)byinsertinganexternalnodesfornon-emptynode

(d)byreplacingeachemptysubtreebyanewexternalnodeUnit–6:SpanningTreesShortquestionanswer

1.Definefollowing. SpanningTree MinimumSpanningTree

2.Drawallthespanningtreesofthisgraph:



3.WhatisthepurposeofSpanningTreeProtocolinaswitchedLAN?

4.WhatisSpanningtree?5.Justifythestatement”Everyconnectedgraphhasaspanningtree.”6.AssumingthatG0hasaminimumspanningT0,isittruethatthenumberofedgesintheT0nogreaterthanT?Answeryesornoandyouranswerexplaininonesentence.7.AssumingthatG’hasaminimumspanningtreeT’,byhowmanyedgescanT’differfromT.8.Howmanyspanningtreedoesthefollowinggraphhave?

9.Howmanyspanningtreedoesthefollowinggraphhave?

10.Howmanyspanningtreesdoesthefollowinggraphhave?

11.Istheminimumweightedgeinagraphalwaysinanyminimumspanningtreeforagraph?Isitalwaysinanysinglesourceshortestpathtreeforthegraph?Justifyyouranswers.12.Drawminimumspanningtree.

13.Statewhetheryourminimumspanningtreeisunique.14.Statedifferencebetweenfollowing:SpanningTreeandminimumspanningtree



Longquestionanswer1.Drawallthespanningtreesofk_3.2.Givethestepbystepprocedureofprim’salgorithmtofindtheminimumspanningtree.3.Givethestepbystepprocedureofkruskal’salgorithmtofindtheminimumspanningtree.4.Foreachofthefollowinggraphs:

(a)Findallspanningtrees.(b)Findallspanningtreesuptoisomorphism.(c)Findalldepth-firstspanningtreesrootedatA.(d)Findalldepth-firstspanningtreesrootedatB.

5.Foreachofthefollowinggraphs:

(a)Findallminimumspanningtrees.

(b)Findallminimumspanningtreesuptoisomorphism.(c)Amongalldepth-firstspanningtreesrootedatA,findthoseofminimumweight.(d)Amongalldepth-firstspanningtreesrootedatB,findthoseofminimumweight.

6.Inthefollowinggraph,theedgesareweightedeither1,2,3,or4.

ReferringtodiscussionfollowingofKruskal’salgorithm:

(a)FindaminimumspanningtreeusingPrim’salgorithm(b)FindaminimumspanningtreeusingKruskal’salgorithm.(c)Findadepth-firstspanningtreerootedatK.

7.LetT=(V,E)beatreeandletd(v)bethedegreeofavertex(a)ProvethatPv2V(2−d(v))=2.(b)Provethat,ifThasavertexofdegreem2,thenithasatleastmverticesofdegree1.(c)Giveanexampleforallm2ofatreewithavertexofdegreemandonlymleaves.

8.Doeseveryconnectedgraphhaveaspanningtree?Eithergiveaprooforacounter-example.9.Giveanalgorithmthatdetermineswhetheragraphhasaspanningtree,andfindssuchatreeifitexists,thattakestimeboundedabovebyapolynomialinvande,wherevisthenumberofvertices,andeisthenumberofedges.



10.Findallspanningtrees(listtheiredgesets)ofthegraph

11.Showthatafinitegraphisconnectedifandonlyifithasaspanningtree.12.Findspanningtreeofmaximal(minimal)weightsforgraphsfromfiguresusingalgorithmsofkruskalandprim.

13.Provethateveryminimalspanningtreeofaconnectedgraphmaybeconstructedbykruskal’salgorithm.14.UseKruskal’salgorithmtofindallleastweightspanningtreesforthisweightedgraph.

15.Findaminimumweightspanningtreeineachofthefollowingweightedgraph

16.Findaminimumweightspanningtreeineachofthefollowingweightedgraph



17.DrawallthespanningtreesofK4. 18.Findthenumberofspanningtreesin

19.

a.UseKruskal’salgorithmtofindaminimalspanningtreeforthegraphbelow.b.UsePrim’salgorithm(asshowninclass)tofindaminimalspanningtreedifferent

fromtheoneinpart

Multiplechoicequestion

1.Howmanyspanningtreesdoesthefollowinggraphhave?

A)1

B)2C)3D)4

Thequestion(s)thatfollowrefertotheproblemoffindingtheminimumspanningtreefortheweightedgraphshownbelow.



3.InFigure,usingKruskal'salgorithm,whichedgeshouldwechoosefirst?

A)ABB)EGC)BED)AG

4.InFigure,usingKruskal'salgorithm,whichedgeshouldwechoosethird?A)EFB)AGC)BGD)EG

5.InFigure,usingKruskal'salgorithm,whichedgeshouldwechooselast?A)ABB)ACC)CDD)BC

6.Infigure,whichofthefollowingedgesofthegivengrapharenotpartoftheminimumspanningtree?

A)ACB)EFC)AGD)BG

7.InFigure,thetotalweightoftheminimumspanningtreeisA)36.B)42.C)55.D)95.

8.WhichofthefollowingstatementsistrueaboutKruskal'salgorithm.A)Itisaninefficientalgorithm,anditnevergivestheminimumspanningtree.B)Itisanefficientalgorithm,anditalwaysgivestheminimumspanningtree.C)Itisanefficientalgorithm,butitdoesn'talwaysgivetheminimumspanningtree.D)Itisaninefficientalgorithm,butitalwaysgivestheminimumspanningtree.

Forthefollowingquestion(s),refertothedigraphbelow.



9.InFigure,whichofthefollowingisnotapathfromvertexEtovertexBinthedigraph?

A)E,B,C,BB)E,D,BC)E,A,BD)E,A,C,B

10.ThecriticalpathalgorithmisA)Anapproximateandinefficientalgorithm.B)Anoptimalandefficientalgorithm.C)Anapproximateandefficientalgorithm.D)Anoptimalandinefficientalgorithm.

11.Letwbetheminimumweightamongalledgeweightsinanundirectedconnectedgraph.Letebeaspecificedgeofweightw.WhichofthefollowingisFALSE? A)Thereisaminimumspanningtreecontaininge.

B)IfeisnotinaminimumspanningtreeTtheninthecycleformedbyaddingetoT,alledgeshavethesameweight.C)Everyminimumspanningtreehasanedgeofweightw.D)eispresentineveryminimumspanningtree.

12.AminimalspanningtreeofagraphGis....?A)AspanningsubgraphB)AtreeC)MinimumweightsD)Allofabove

13.Atreehavingamainnode,whichhasnopredecessoris....?A)SpanningtreeB)RootedtreeB)WeightedtreeD)Noneofthese

14.IfagraphisatreethenA)ithas2spanningtreesB)ithasonly1spanningtreeC)ithas4spanningtreesD)ithas5spanningtrees

15.AnytwospanningtreesforagraphA)DoesnotcontainsamenumberofedgesB)HavethesamedegreeofcorrespondingedgesC)containsamenumberofedgesD)Mayormaynotcontainsamenumberofedges

16.AsubgraphofagraphGthatcontainseveryvertexofGandisatreeiscalledA)Trivialtree B)emptytree



C)Spanningtree D)MinimumSpanningtree 17.Theminimumnumberofspanningtreesinaconnectedgraphwith“n”nodesis….. A)n-1 B)n/2 C)2 D)1 18.AminimalspanningtreeofagraphGis

A)Aspanningsubgraph B)Atree C)Minimumweights D)Allofabove

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