MC0063(B)-unit13-fi

ModernApplicationsusingDiscreteMathematicalStructures Unit13

SikkimManipalUniversity PageNo:286

Unit13 DirectedGraphsStructure

13.1 Introduction

Objectives

13.2 DefinitionsandExamples

13.3 Binaryrelationasadigraph

13.4 EulersDigraphs

13.5 MatrixRepresentationofDigraphs

SelfAssessmentQuestions

13.7 Summary

13.8 TerminalQuestions

13.9 Answers

13.1Introduction

Thegraphssofarstudiedareundirectedgraphs(nodirectionwasassignedto

theedgeinagraph).Inthisunitweshallconsiderdirectedgraphs(graphsin

whichedgeshavedirections).Manyphysicalsituationsrequiredirectedgraphs.

Theapplicationsofdirectedgraphsareinthestreetmapofcitywithoneway

streets, flow networks with valves in the pipes, and electrical networks.

Directedgraphsintheformofsignalflowgraphsareusedforsystemanalysis

incontroltheory.Mostoftheconceptsandterminologyofundirectedgraphs

are also applicable to directed graphs. In this unit we see the some of the

propertiesofdirectedgraphsarenotsharedbyundirectedgraphs.

Objectives

Attheendoftheunitthestudentmustbeableto:

i) Writetheindegreeandoutdegreeofavertex.

ii) Observethedifferentmatrixrepresentationsofdigraphs.

iii) KnowthetournamentsandEulersdigraphs..

iv) Applythepropertiestoflows,networkandtrafficproblem.



13.2DefinitionsandExamples

13.2.1Definition:

A directed graph (or) a digraph D consists of a nonempty set V

(theelementsof V arenormallydenotedby v1,v2,)andaset E(the

elementsof Earenormallydenotedby e1,e2,.)andamapping y that

mapseveryelementof E ontoanorderedpair(vi,vj)ofelementsfrom V.

Theelementsof V arecalledasverticesornodesorpoints. Theelements

of E arecalledasedgesorarcsorlines.If e E and vi,vj V such

that y (e)=(vi,vj),thenwewrite e = jivv .Inthiscase,wesaythat e

isanedgebetween vi and vj.(Wealsosaythat e isanedgefrom vi to

vj).(Wealsosaythat e originatesat vi andterminatesat vj).(Anedge

from vi to vj isdenotedbyalinesegmentwithanarrowdirectedfrom vi

to vj).

13.2.2Note

i) Adirectedgraphisalsorefereedtoasanorientedgraph.

ii) LetD beadirectedgraphande =vu .

Then we say that e is incident out of the vertex v, and incident into the

vertex u.Inthiscase,wesaythatthevertex v iscalledtheinitialvertex

andthevertex u iscalledtheterminalvertexofe.

13.2.3 Example

ThegraphgiveninFigure13.2.3isadigraphwith5verticesandtenedges.

e9

v2

e10

e5

e7

v1

e8e6

v4

v3

v5

e1e4

e3 e2 Fig .13.2.3



Here v5 istheinitialvertexand v4 istheterminalvertexfortheedgee7.

Theedgee5 isaselfloop.

13.2.4Definition

i) The number of edges incident out of a vertex v is called the out

degree(or)outvalenceof v.Theoutdegreeofavertex v isdenoted

by d+(v).

ii) The number of edges incident into v is called the indegree (or) in

valanceof v. The indegreeofavertex v isdenotedby d(v).Note

thatthedegreeof v isequaltothesumofindegreeandoutdegreeof

v, foranyvertex v inagraph.(Insymbols, wecanwriteasd(v)=

d+(v)+d(v)forallvertices v).

13.2.5Example

ConsiderthegraphgiveninFig.13.2.3.

i) Here d+(v1)=3, d+(v2)=1, d+(v3)=1, d+(v4)=1, d+(v5)=4. d(v1)=1,

d(v2)=2, d(v3)=4,d(v4)=3, d(v5)=0.

ii) d(v1)=4=3+1=d+(v1)+d(v1) d(v2)=3=1+2=d+(v2)+d(v2),

andsoon.

13.2.6 Problem

Let D be a directed graph. Show that the sum of all indegrees of the

verticesof D, isequal tothesumofalloutdegreesof theverticesof D

andeachsumisequaltothenumberofedgesin D.

[Inotherwords, =

+ n

1ii )v(d =

=

- n

1ii )v(d =qwhereqisthenumber

ofedgesin D,and{v1,v2,,vn}isthesetofallverticesin D].

Solution:Let D beanydigraphwith n verticesandqedges.Whenthe

indegreesofverticesarecounted,eachedgeiscountedexactlyonce.

[Thisis,becauseeveryedgegoestoexactlyonevertex].



So =

- n

1ii )v(d =q ...(i)

Similarly, when the outdegrees of vertices are counted, each edge is

countedexactlyonce.[Thisis,becauseeachedgegoesoutofexactlyone

vertex].

So =

+ n

1ii )v(d =q .(ii).

From(i)and(ii),wegetthat =

+ n

1ii )v(d =

=

- n

1ii )v(d =q.

13.2.7Example

Consider thedigraph inFig.13.2.3.Here thenumberofvertices n = 5,

andthenumberofedgesq=10.

Now =

+ n

1ii )v(d = d

+(v1)+d+(v2)+d+(v3)+d+(v4)+d+(v5)=3+1+1+1

+4=10=q

Also =

- n

1ii )v(d = d

(v1)+ d(v2)+d(v3)+d(v4)+d(v5)=1+2+4+

3+0=10=q.

Therefore =

+ n

1ii )v(d =

=

- n

1ii )v(d =q=thenumberofedges.

13.2.8Definitions

i) Avertexvissaidtobeanisolatedvertexif theoutdegreeof v and

theindegreeof v areequaltozero.(Insymbols,d+(v)=0=d(v)).

ii) Avertex v inadigraph D issaid tobeapendentvertex if it isof

degree 1. (Inotherwords,avertex issaid tobeapendentvertex if

thedegreeof v = d+(v)+d(v)=1).



iii) Twodirectededgesaresaid tobeparalleledges if theyaremapped

ontothesameorderedpairofvertices.

iv) In other words, two directed edges e and f are said to be parallel

edges if both e and f originates at the same vertex, and also

terminatesatthesamevertex.(InthegraphgiveninFig.13.2.3,the

edges e8, e9,ande10 areparalleledges.Theedgese2 and e3 are

notparallel).

13.2.9Note

i) Let D be directed graph. If we disregard the orientation (that is,

direction)ofeveryedgein D, thenwegetanundirectedgraph.The

undirectedgraphobtained(inthisway)from D iscalledtheundirected

graphcorrespondingto D.

ii) SupposeHisanundirectedgraph.ToeachedgeofHwecanassign

adirection.ThenthedigraphobtainediscalledanorientationofH.(or

adirectedgraphassociatedwithH).

13.2.10Definition

Let D1and D2betwodigraphs.Thesetwographsaresaid tobe isomorphic if

i) thecorrespondingundirectedgraphsareisomorphic,and

ii) thedirectionsofcorrespondingedgesmustbesame.

13.2.11Example

i) Consider the directed graphs D1 and D2 given. The two digraphs D1

and D2 arenot isomorphic(becausethedirectionof theedgee4 in

D1 isdifferentfromthedirectionoftheedgee4 in D2).

e1 e4

e5

e2

e3

e6

e1 e4

e5

e2

e3

e6GraphD1 GraphD2



ii) Note that the undirected graphs corresponding to D1 and D2 are

isomorphic.

13.2.12Definition

Adigraph that hasneither selfloopsnorparallel edges is calleda simple

digraph.

Inotherwords, adirectedgraph D is said tobea simpledigraph if the

correspondingundirectedgraphisasimplegraph.Thetwodigraphsgiven

inExample13.2.11aresimpledirectedgraphs.

13.2.13Definition

i) Digraphs that have at most one directed edge between any pair of

vertices,butareallowed tohaveselfloopsarecalled theasymmetric

(or)antisymmetricdigraphs.

ii) AdigraphD issaidtobeasymmetricdigraphifforeveryedge(a,b)

in D thereisalsoanedge(b,a)in D.

iii) A digraph that is both simple and symmetric is called a simple

symmetricdigraph.

iv) A digraph that is both simple and asymmetric is called simple

asymmetricdigraph.

v) A simple digraph is said to be a complete symmetric digraph if it

satisfiesthefollowingcondition:"Givenanytwoverticesv1andv2 there

correspondexactlyoneedgedirectedfrom v1 to v2".

vi) A complete asymmetric digraph (or) tournament (or) a complete

tournament. is an asymmetric digraph in which there is exactly one

edgebetweeneverypairofvertices.

13.2.14Example

Inthefollowing,

i) ThegraphD1 isnotasymmetric since thereexist twodirectededges

between v1 and v2.



ii) ThegraphD2 isasymmetric.

iii) Anynullgraphisasymmetric.

iv) ThedigraphD3 isacompletesymmetricdigraph.

13.2.15Observation:

i) A tournament (means, complete asymmetric digraph) of n vertices

contains ( ) 2

1nn - edges.

Thedirectedgraphgivenin13.2.11isatournament.Inthisgraph,the

numberofverticesis n=4.

Thenumberedgesis6= ( ) 2

144 - =

( ) 2

1nn - .

ii) Acompletesymmetricgraphof n verticescontains n(n1) edges.

The digraph given in 13.2.14, D3 is a complete symmetric graph on

n =4vertices.

Inthisgraph,thenumberofedgesis12=4(41)= n(n1).

13.2.16Definition

Adigraph D issaidtobebalancedif the indegreeof v isequal totheout

degreeof v foreveryvertex v in D.Inotherwords,adigraph D issaidto

bebalancedifd+(v)= d(v)forallvertices v inthedigraph D.Abalanced

digraph isalsoknownas pseudosymmetricdigraph(or)anisograph.

v2v3

v1

G1 G2

G3



Abalanceddigraphissaidtobearegulardigraph if itsatisfythefollowing

twoconditions:

i) Theindegreesofalltheverticesof D areequaland

ii) Theoutdegreesofalltheverticesof D areequal.

13.3Binaryrelationasadigraph

Let X beaset.Representtheelementsof X bythesymbols x1,x2,

Supposethat R isarelationon X.Werepresenttheelementsof X as

verticesanddrawadirectededgefrom xi to xj if(xi,xj) R.Thenweget

adirectedgraphwhichrepresentsthegivenrelation R on X.

13.3.1Example:

Considertheset X={3,4,5,7,8}andtherelation(R,>)on X.

Then R={(8,3),(8,4),(8,5),(8,7),(7,3),(7,4),(7,5),(5,4),(4,3)}

={(x,y)/ x>y, x,y X}.

Thegraph D giveninbelowrepresentsthisrelation R on X.

Itisclearthat,everybinaryrelationonafinitesetcanberepresentedbya

directedgraphwithout parallel edges. Conversely, if a directed graph D

withoutparalleledgeswasgiven,thentherecorrespondsabinaryrelation

onthesetofvertices.

57

3

4

8



13.3.2Definition:

i) Let X beaset,and R arelationon X(thatis, R X X).If(x,y)

R, thenwealsowrite xRy.Therelation R issaidtobereflexive

relation ifxRx forall x X.

(Notethatthedigraphsofareflexiverelationhaveaselfloopatevery

vertex).

Wecalladirectedgraphrepresentingareflexivebinaryrelationonits

vertexsetasreflexivedigraph.

A digraph in which no vertex has a selfloop is called an irreflexive

digraph.

ii) Let X beaset,and R arelationon X.Wesaythat R is symmetric

if a,b X, aRb bRa. Notethatadirectedgraph representinga

symmetricrelationisasymmetricdigraph.

13.3.3 Example

ThegraphgiveninD1 isthedigraphofarelationwhichissymmetricbutnot

reflexive.Thisrelationisontheset{x1,x2,x3,x4}.Someauthorsrepresent

thisrelationbytheundirectedgraphgiveninD2.NotethatinthegraphD2

onlyoneundirectededgetakenbetweenapairofvertices(thatarerelated

undertherelationR).

13.3.4Definition:

Let X beaset, and R a relationon X.The relation R issaidtobe

transitive if a, b, c R, aRb, bRc aRc. A digraph representing a

transitive relation is called a transitive digraph. [Observe that the graph

x1 x2

x3 x4

GraphG1

x1 x2

x4 x3

GraphG2



givenin13.3.1isatransitivedigraph].

Let X beaset,and R arelationon X.

i) The relation R is said to be an equivalence relation if it is reflexive,

symmetricandtransitive.

ii) Adigraphrepresentinganequivalencerelationiscalledanequivalence

digraph.

13.3.5Example

ConsiderthebinaryrelationR(=Congruentmodulo3)definedontheset

X={10,11,12,13,14,15,16,17,18,19,20}.

Wecanobservethatthisrelation Ron X isanequivalencerelation.The

related equivalencegraphwas given.We are use undirected graph here.

Observe that the set of vertices is divided into three disjoint equivalence

classes.Eachofthesesetsformseparatecomponents.Eachcomponentis

anundirectedsubgraph(duetosymmetry, weareusingundirectedgraph

withaselfloopateachvertex).Alsonotethat(sinceanytwoelementsinan

equivalenceclassarerelated)anytwovertices insidethecomponentwere

joinedbyanedge.

13.3.6Definition

LetXbeaset,andRisabinaryrelationonX.ThisrelationRonXmaybe

representedbyamatrix.Thismatrix iscalledasrelationmatrix. It isa (0,1)

nnmatrix,where n isthenumberofelementsintheset X ={x1,x2,, xn}.

11 14

17 20

)3(mod2

10 13

16 19

)3(mod1

15

12

18

)3(mod0



Therelationmatrix( ija )isdefinedasfollows:

ija =1if xiRxj.

=0otherwise.

13.3.7 Example:

The relation matrix of the relation R (" usual grater than") on the set

X={3,4,5,7,8},istheMatrix

13.3.8Definition:

i) Apath v0e1v1e2v2envn issaidtobea directedpathifek isoriented

from vk1 to vk foreach1 k n.Apathwhichisnotadirected

pathiscalledasemipath.Inadirectedgraph,theword"path"refers

toeitheradirectedpathorasemipath.

ii) Let D beadirectedgraph.Adirectedwalk(in D)fromavertex v to

u isanalternatingsequenceofverticesandedgesbeginningwith v

andendingwith u such that eachoneof theedges isoriented (or

directed)fromthevertexprecedingittothevertexfollowingit.[Inother

words,awalk v0e1v1e2v2envn intheundirectedgraph D issaidto

beadirectedwalkif ek isorientedfrom vk1 to vk forall1 k n].

13.3.9Example:

ConsiderthegraphgivenintheFig13.2.3.

i) Thepath v5e8v3e6v4e3v1 isadirectedpathfrom v5 tov1.

ii) Ifwedisregardtheorientationin D,thenv5e7 v4e6 v3e1v1 isapath.

Observethatitisnotadirectedpath.Soitisa"semipath".

34758

34

7

5

8

01111

00011

01011

00001

00000

.



Sinceadirectedwalkisawalk,wehavethatnoedgeappearsmorethan

once(butavertexmayappearmorethanonce).

13.3.10Definition:

Let D beadigraph.

i) Awalkwhichisnotadirectedwalkiscalledassemiwalk.Weusethe

term"walk"tomeaneitheradirectedwalk(or)asemiwalk.

ii) Acircuit(inthecorrespondingundirectedgraph)v0e1v1e2v2envn issaid

tobea directedcircuit if ek isorientedfrom vk1 to vk forall1 k n.

iii) Acircuitwhichisnotadirectedcircuitisasemicircuit.

13.3.11Example:

Considerthedigraph D given13.2.3.

i) Thecircuit v1e1v3e6v4e3v1 isadirectedcircuit.

ii) Thecircuit v1e1v3e6v4e2v1 isnotadirectedcircuit.Soitisasemicircuit.

Inundirectedgraphs,wedefinetheconnectednessbyusingthenotion"path".

Astherearetwodifferenttypesofpaths(namelydirectedpath,semipath) in

digraphs,wehavetwodifferenttypesofconnectednessinthedigraphs.

13.3.12Definition

Let D beadigraph.

i) D issaidtobestronglyconnectedifthereexistsatleastonedirected

pathfromeveryvertextoeveryothervertex.

ii) D issaidtobeweaklyconnectedifitscorrespondingundirectedgraph

isconnected,but D isnotstronglyconnected.

iii) Wesaythat D isconnectediftheundirectedgraphcorrespondingto

D is connected. So the word "connected digraph" refers to both

stronglyconnecteddigraphandweaklyconnecteddigraph.

iv) If D isnotconnected,thenwesaythat D isadisconnectedgraph.

13.3.13Example:

ConsiderthedigraphsD1 and D2 giveninthefollowing.

i) ThegraphD1 isastronglyconnectedgraph.



ii) In the graphD2, there is no directed path from v1 to v4 so it is a

weaklyconnecteddigraph.

Since there are two types of connectedness in a digraph, we define two

typesofcomponents.

13.3.14Definition

Let D beadirectedgraph.

i) Eachmaximal connected (weakly or strongly) subgraph of a digraph

D,issaidtobeacomponentof D.

ii) With in each component of D, the maximal strongly connected

subgraphs are said to be the fragments (or) strongly connected

fragmentsof D.

13.3.15Example:

ConsiderthedigraphgivenintheFig13.3.15.

ThisgraphconsistsoftwocomponentsH1andH2.

e2 e2

v4

v1

e1 e4

e5

e3

e6

e1 e4

e5

e3

e6GraphD1 GraphD2

v1

e9

e13

e12

e14

e11

Fig13.3.15

e10

e1

e2

e5

e3 e4

e6e7

e8

H1

g2



i) ThecomponentH1 containsthreefragments:{e1,e2}{e5, e6, e7, e8}

and{e10}.

ii) ThecomponentH2 containsafragment {e11, e12, e13}.

iii) Itcanbeobservedthat e3, e4 and e9 donotappearinanyfragment

ofH1.

13.3.6Definition:

Let Dbeadigraph.Thecondensation Dc of D isadigraphobtainedby

usingthefollowingmethod:

i) Eachstronglyconnectedfragmentistobereplacedbyanewvertex,and

ii) Foranytwostronglyconnectedcomponentsc1 andc2,thesetofall

thedirectededgesfromc1 toc2 istobereplacedbyasingledirected

edge(fromc1 toc2).

iii) Let D beadirectedgraphand v, u areverticesin D.Then v said

tobeaccessible(or)reachablefromthevertex u,ifthereisadirected

pathfrom u to v.

13.3.7Example:

Considerthegraph D givenintheFig13.3.15.

i) Thecomponentsof D areg1,g2.

ii) Thefragmentsof D areC1={e1,e2}

C2={e5,e6,e7,e8}C3={e10}andC4={e11,e12,e13}.

iii) ThefragmentsC1,C2,C3 andC4 aretobeconsideredasvertices.

iv) Therearetwoedges e3 and e4 fromC1 toC2.

Thesetwoedges e3 and e4 aretobereplacedbyanedge(sayf1)

fromtheverticesC1 toC2.

v) Thereisonlyoneedge e9 fromthefragmentC2 toC3.Sothisedge

e9 istobedrawnfromthevertexC2 toC3.

vi)Thereisonlyoneedge(e14)fromthefragmentC4 tothevertexv1. So

thisedgeistobedrawn(inthecondensation)fromthevertexC4 to v1.

vii)Finallywegetthecondensation Dc thatwasgivenintheFig13.3.7.



13.3.8Observations

i) Thecondensation Dc ofastronglyconnecteddigraph D isavertex.

ii) Thecondensation Dc ofadigraph D containsnodirectedcircuits.

Itisclearthatadigraph D isstronglyconnected "v isaccessiblefrom

u"forallvertices v and u in D.

13.4EulersDigraphs

13.4.1Definition:

LetDbeadirectedgraph.

i) A directed walk that starts and ends at the same vertex is called a

closeddirectedwalk.

ii) Acloseddirectedwalkwhichtraverseseveryedgeof D exactlyonce,

iscalledadirectedEulerline.

iii) D issaidtobeanEulerdigraphifitcontainsadirectedEulerline.

13.4.2 Example:

Considerthegraphgiveninthefollowing.

e9

v1 Fig. 13.3.7

C4

e14

C3

C2

f1

C1

a

fc

e

c

1

2

b

dc

4

3



Theedgesequencef,a,b,c,d,e formsadirectedEuler line.Hencethis

graphisaEulerdigraph.

13.4.3Theorem:

Let D be a digraph. Then D is an Euler digraph if and only if D is

connectedandbalanced.

[thatis, d+(v)=d(v)foreveryvertex v in D].

13.4.4Definition:

A connected digraph containingno circuit (neither a directed circuit nor a

semicircuit)issaidtobeatree.

13.4.5Note:

i) Atreeon n verticescontain n1edges(directed).

ii) Treesindigraphshaveadditionalproperties(thanthoseinundirected

graphs) and variations resulting from the relative orientations of the

edges.

13.4.6Definition:

A digraph D is said to be an arborescence if it satisfy the following two

conditions:

i) Dcontainsnocircuit(neitheradirectedcircuitnorasemicircuit).

ii) Thereexistsexactlyonevertex v ofzeroindegree(thisvertex v is

calledtherootofthearborescence).

13.4.7 Example:

Thegraphgivenhereisarborescence.

v(root)



13.5 MatrixRepresentationofDigraphs

13.5.1Definition:

Let D be a digraph with p vertices. The adjacencymatrix of D is a pp

matrix(aij)with

aij= i j1if v v isanarcof D

0otherwise

.ItisdenotedbyA(D).

Theadjacencymatrixforthedigraphgiveninfigure13.5.1is

01110

01000

00010

10000

00110

.

Thesumof the ith rowentriesof theadjacencymatrixgivesd+(vi)and the

sumoftheithcolumnentriesgivesd(vi)foreveryi.

ThepowersofA(D)givethenumberofwalksfromonepointtoanother.

13.5.2Problem:

The(i,j)thentryAn isthenumberofwalksoflengthnfromvi tovj.

Solution:(Byinductiononn)

Step1:n=1thestatementistruebythedefinitionofadjacencymatrix.

Step2:Inductionhypothesis:thetheoremistrueforn1.

LetAn1=(bij).Thereforebij=numberofvivjwalksoflengthn1.

v2

e5

e7

v1

e8

e6

v4

v3

v5

e1

e4

e3e2

fig13.5.1



Step3:ConsiderAn=An1A.

Thereforethe(i,j)th entryofAn=bi1a1j++bipapj.(1)

BythedefinitionofA(D),akj=numberofvkvjwalksoflength1.

Hence by step 2, for every k, 1 k p, bikakj = number of vivj walks of

lengthnwhoselastarcisvkvj.

Since anywalkhas one among v1vj, v2vj,,vpvj as the last arc, the right

handsideofexpression(1)givesthenumberofvivjwalksoflengthn.

Hence(i,j)thentryofAn isthenumberofvivjwalksoflengthn.

13.5.3Definition:

i) LetDbedigraphwithpvertices.ThereachabilitymatrixR=(rij)isthe

ppmatrixwith

j iij

1if v isreachablefromvr =

0otherwise

. We assume that each vertex is

reachablefromitself.

ii) The distance matrix is the pp matrix whose (i,j)th entry gives the

distancefromthepointvi tothepointvjandisinfinityifthereisnopath

fromvi tovj.

iii) Thedetourmatrix is theppmatrixwhose (i,j)thentry is the lengthof

anylongestvivjpathandinfinityifthereisnosuchpath.

13.5.4Example:

Thereachablematrix,distancematrixandthedetourmatrixrespectivelyare

forthematrixgivenin13.5.1aregivenbelow.

11110 111 112

01000

10110 , 2331 and 2331 .

11110 1222 1223

11110 2331 2332

1 1




1.Findtheindegreeandoutdegreeofthefollowingdigraphs

2. Findthespanningtreeofthedigraphgivenin1(ii).

3. Verifywhetherornotthefollowingdigraphsareisomorphic?Ifso,write

theisomorphisms.

13.7Summary

In this unit we have investigated the fundamental featuresof the directed

graphs.Onecanobservethatthepropertiesofdigraphsaresimilartosome

typesofundirectedgraphs.Thecloserelationshipbetweenbinaryrelation

anddigraphswasgiven. Directedgraphshave interestingapplications in

telecommunications,flowsandnetworksandroutingproblems.

(i)

c

ba

o

oo

c

a

b

d

e

(ii)

o

o

oo

o

1

2

3

4o

o

o

o

G:

a

b

cd

G1:

o o

o

o



13.8TerminalQuestions

1. Draw thedigraphs for thedigraphD = {a, b, c, d, e}where thearcs

representedby {(a,c), (a,d), (b,e), (e,c), (d,c)}. Write the

indegreeandoudegreeofD.

FindalsotheconverseD1 (i.e.,reversingtheeacharcdirection inD)

andfindtheindegreeandoutdegreeofeachpointinD1.

2. Giveanexampleofadirectedtreewithfivevertices.

3. Findindegreeandoutdegreeofeachvertex inthefollowingdirected

graphs.

13.9Answers


1. i) Indegreea=1

outdegreea=2

Similarlyid(b)=1,Od(b)=1

id(c)=2,Od(c)=1

ii) id(a)=1id(b)=2id(c)=3id(d)=1id(e)=1

Od(a)=1Od(b)=2Od(c)=2Od(d)=1Od(e)=1

(i)

v1 v2

v3v4

v5

o

o

o

oo

(ii)

d

a b

c

oo

oo



2.

3. DandD1areisomorphicundertheisomorphismf(1)=a,f(2)=b,f(3)=

c,f(4)=d.

TerminalQuestions

1 Thedigraphrepresentingthegivenarcsis

Theconversedigraphalsocanbefoundinasimilarway.

2.

a

bc

e

d

aa

o

oo

o

o o

c

ebo

o

o o

ob

d

a

d

e

ca bo

o

o

o o



3. i) indegreev1=2,outdegreev1=1

indegreev2=2,outdegreev2=2

Similarforothervertices.(ii).Similarto(i).



SuggestedReferences

1. Bernard Kolman, R.C. Busby, Sharon Ross, Discrete Mathematical

StructuresPHI,1999.

2. Deo Narsing, Graph Theory with Applications to Engineering and

ComputerScience,PrenticeHallIndia,NewDelhi1986.

3. FraleighJ.B.AFirstCourseinAbstractAlgebra,NarosaPub.House

,NewDelhi,1992

4. Graham R. L., Knuth, D. E., and Patashnik O., Concrete

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Wesley,1994.

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