Curriculum: From Theory to Practice 2nd Edition
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Graphs, Orders, Infinites and Philosophy
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Chapter1.Graphs .....................................1
1.1.Graphtheory:abriefhistory.............................1
1.2.Basicdefinitions...................................6
1.3.Differenttypesofgraphs...............................6
1.4.Moreonthelistofgraphs..............................9
1.5.Graphsandvertices..................................11
1.6.Someoperationsongraphs.............................13
1.7.Graphisomorphisms.................................15
1.7.1.Self-complementarygraphs...........................15
1.7.2.Propertiesofself-complementarygraphs...................16
1.8.Symmetricandasymmetricgraphs.........................17
1.9.Extremalgraphs...................................20
1.10.Independence,non-separability,reconstructionconjecture. ..........23
2.1.Ancientmappings..................................29
2.2.Chinesetetragrams..................................33
2.3.Pythagorismandpentagram.............................33
2.4. n-gramsandsomefiguresoftheworld.......................35
2.5.Graphsandclassicalsystematicity.........................41
2.6.Towardsanewkindofsystematicity........................52
2.6.1.Non-Hamiltonianandnon-Eulerianphilosophies ...............52
2.6.2.PancyclicgraphsandMetahegelianism....................57
2.7.Non-pythagorismandarrangementoflines....................58
2.7.1.Levigraphsoflinearrangements........................62
2.7.2.Linearrangementsofcurvelines........................65
2.7.3.Hyperbolicgraphs................................67
Chapter3.OrderandItsPhilosophicalUse
3.1.Themathematicalnotionoforder:abriefhistory.................74
3.2.Theideaof“well-ordering”.............................76
3.3.Quasiorders(orpreorders)..............................79
3.4.Partialorders.....................................81
3.4.1.Thenotionofwellpartialorder.........................84
3.4.2.Linearextensionofaposet...........................84
3.4.3.Wellpartiallyorderings.............................85
3.5.Trees..........................................85
3.5.1.Azooofinfinitetrees..............................86
3.5.2.Ordinalinfiniteclassifications.........................87
3.6.Moralproblemsinafiniteworld..........................87
3.7.Orderversuscircularity...............................92
3.8.Conclusion......................................96
Chapter4.TowardsaFormalPhilosophy
4.1.Asenjo’ssystemsandDubarle’sformalizationofHegelianism..........99
4.1.1.Asenjo’ssystemsandDubarle’scase......................100
4.1.2.Dubarle,Parmenides’thoughtandHegel ...................102
4.1.3.Projectivealgebras................................103
4.2.Somecriticisms ....................................106
4.3.Porphyryandtheneoplatonistmodeofthought... ...............107
4.4.AvariantofDubarle’sformalism..........................109
4.5.Quasi-Hegeliansystems...............................112
4.6.Philosophicalthinkingandfiniteprojectivegeometry ...............114
4.7.Otheralgebrasforphilosophicalthinking. ....................116
4.8.Modelsderivedfromgeometryandalgebraicgeometry..............117
4.9.Conclusion......................................120
Chapter5.PhilosophicalTransformations
5.1.Theparadoxofametasystem............................121
5.2.Insearchofanalgebra................................137
Chapter6.ConceptsandTopology ..........................141
6.1.Formalconcepts...................................141
6.2.Fuzzyconcepts....................................143
6.3.Thecaseofphilosophicalconcepts .........................144
Chapter7.TheProblemoftheInfinite ........................155
7.1.Thearithmeticofinfinitecardinals.........................156
7.2.Thequestionoflargecardinals...........................158
7.3.Woodin’sprogram... ...............................163
7.3.1.WoodinI:CHwouldbefalse. .........................163
7.3.2.WoodinII:CHmaybetrue.. .........................165
7.3.3.Ultimate L andCH...............................167
7.4.Infiniteandphilosophy. ...............................167
Chapter8.InSearchforaNewPhilosophy .....................171
8.1.Thefinitecase.....................................171
8.2.Theinfinitecase...................................176
Chapter9.ExtensionofStructuralismandNegativeTheology ........181
9.1.Complementaritygraphs...............................182
9.2.Orderrelation,orderedset..............................184
9.3.Graphsassociatedwithapartiallyorderedset...................185
9.4.Complementarityandincomparabilitygraphsofaposet... ..........187
9.5.Booleanrepresentationofaposet..........................187
9.6.Caseoflattices....................................189
9.6.1.CaseofBooleanlattices.............................191
9.6.2.Generalization:Booleanlatticesas n-cubes..................193
9.7.Consequencesfornegativetheology........................196
Chapter10.FromFuzzyGraphstoNeutrosophicGraphs ...........201
10.1.Fuzzysets......................................201
10.2.Fuzzygraphs....................................204
10.3.Intuitionisticfuzzysettheory...........................206
10.4.Neutrosophy. ....................................208
10.5.Single-valuedneutrosophicsetsandgraphs ....................211
Infact,mathematicsisoneofthemainsourcesofintelligibilityofreality. Butitsroledoesnotstopthere.Itcanalsoprovidephilosophywithmoreprecise conceptsthantheonesweordinarilyusetointerpretthesensibleworld.Aswe shallsee,discoveriesinmathematicssometimeslimitordisputetheclaimsofpast doctrines,buttheymayalsosuggestnewwaysofposingproblems.
Wefirstwishtoillustratetheseaffirmationsbysomereflectionsongraph’s problems,asthemathematicianstodayconceivethem.Wewillgoonby someconsiderationsonorder,pre-orderandpartialorder,thencarryoutmore geometricandtopologicalinvestigations.Intheend,wetacklethequestionof infinityandcometopossibleapplicationsinourdomainofsomeparticular graphs(incomparability,fuzzy,neutrosophicorpolyhedral graphs).Weonlyuse commonnotationsandremain,themostofthetime,asclearaspossible,in ordertogivethephilosophicalreadersomeusefultoolsfordevelopingtheirown thought.
Letusnowdefinewhata“philosophicalsystem”isforus,i.e.the“unit”on whichphilosophicalconsiderationsarebased.Iemphasizethispoint:thehistory ofthought,thesociologyofknowledgemaybeconcernedwithphilosophical ideas.Buttheproperhistoryofphilosophyisconcernedwith systems,or,atleast, philosophicalconstructionswhichmustexplicitlyexplainwhytheyarenot–or cannotbe–systems.
Definition: Wewillcalla philosophicalsystem atheory,expressedinthe commonlanguageorasaturatedextensionofit,whichtriestogiveaconceptual imageofthewholeworld,oreventuallyexplainswhysuchanimageisimpossible toconstruct.
Commentary: Thesentence“expressedinthecommonlanguageorasaturated extensionofit”allowsustodistinguishbetweenphilosophyandscience. Generally,philosophyiswritteninalanguageveryclosetothenaturallanguage andthatmaybereadbyeveryone.Forexample,thisisthecaseinPlato’s dialogues,intheworksofFrenchauthorsfromthe18thcenturysuchasRousseau andDiderot,orintheessaybyFichteentitled Thedestinationofman.However, inamoreelaboratedform,philosophycanmakeuseofmoreorlessabstract conceptsandsometimesreferstoscientificlanguage(see,forinstance,the LogicalInvestigations byHusserl).Theexpression“saturatedextension”comes fromlogicandwasintroducedbyJeanLadrière(seeLadrière1972) 2 .
2Tokeepitsimpleandnotgetintotootechnicalconsiderationsofmodeltheory,wecansaythat alogicalsystemis saturated ifanyadditionofasupplementary thesismakesitcontradictory or,ifittoleratescertainformsofcontradictioninitself,inconsistentwithrespecttoaparticular operationdifferentfromclassicalnegation.
Moreover,inordertocompletethisdistinctionbetweenscienceand philosophy,theterm“world”hastobeinterpretedinalargersensethanthe term“universe”.Thenwecanseparatephilosophyfromthisbranchofphysics called“cosmology”.Classicalphilosophyisakindofexistentialcosmology supportingthenon-physicaldimensionsoftheworld.Foraclassicalphilosopher, theuniverseor,afortiori,theobserveduniverse,isonlyasmallpartofalarger entitythattheGerman-SwissphilosopherKarlJaspers(1883–1969)named“the Encompassing” 3 .
Inthiscontext,theclassicphilosophicaloperationconsistsofpostulatingthat thereexistsamapping:
φ : W → S
betweenametaphysicalentity W ,whichissupposedtorepresentthewholeworld initscomplexity 4,andaphilosophicalsystem S ,whichcapturesitsmainfeatures.
Whenaphilosopherthinksthatthisoperationisimpossible,thenthey generallytrytoshowthat,insomeparticularsystem SP writtenbyaphilosopher P ,thereexistssomelinguisticexpression x towhich P didnotgiveanysense,so wecannotexhibitanycorrespondingconcreteentityin W .Insymbols:
Commentary: FollowingthesuggestionsofHume,Kantwasprobablythe firstphilosopherwhoappliesthismethod.Heprovesthatwecannothaveany knowledgeofentitiesaboutwhichwegetnoinformation(inhislanguage,no empiricalorpureintuitions),sothatthethreegreatmetaphysical“objects”named God,WorldandSoulhavenocorrespondingrealityintheconcreteexperience. Wittgensteinpursuethistaskinhis Tractatuslogico-philosophicus andbeganto huntdownallthelinguisticexpressionsthatphilosophersusedinanaivewayand thatwereinthesamesituation.Thisproject,inamoreorlessstrongform,was thereaftercontinuedbyallso-called analyticalphilosophy.
3Becauseofthesubject–objectsplit,whichalwaysplacesourconsciousnessoutsidetheworld ofobjects,beingasawholecanbeneitherobjectnorsubject.AccordingtoJaspers,itmustbe the“encompassing”thatmanifestsinthissplit.Objectsassubjectsthereforeariseagainstthe backgroundofthisencompassing,whichremainspartlyobscure.Itisclarifiedonlybyobjects, anditbecomesallthemoreclearastheobjectsaremoreclearlypresenttoconsciousness.But, forallthat,itdoesnotitselfbecomeanobject.Itisthereforebasicallywhat,throughthought, onlyannouncesitself.Wenevermeetititself,buteverythingwedomeet,wemeetinit.
4Thiscomplexityisassumed,butitisnotyetknownwhenthephilosopherbeginstheirwork. Inordertojustifythepossibilityofthisoperation,somephilosopherslikeM.Gueroulthave spokenofthisentity W asa“commonreal”(seeGueroult1979).Othershavesimplypostulated thatthereis,ineveryone,a“thoughtofaframeofreference”(see,forexample,Granier1977).
Inthisbook,wewanttoexploretheactualpossibilityofconstructingtoday’s philosophiesbystudyingthedifferentformsoforganizationsthattheycantake. Wewillshowthatmathematicsgivesalotoftoolstobuildrobustconceptual architecturessothatwemayhopethatsystematicphilosophyisalwayspossible 5
Thisundoubtedlyrestsonseveralpostulateswhichcanbelistedasfollows:
–(P1 )Itispossibletorepresentwithoutbias,inasingleformandoflimited dimension–whatiscalleda“philosophicalsystem”–theessentialofwhat constitutesourexperienceoftheworld.
–(P2 )Aphilosophicalsystemisusefulandevennecessary(ifnot,whywaste timebuildingit?).
–(P3 )Therelationalstructure(generallyexpressedasagraph 6)ofa philosophicalsystemisconnectedinsomesensetotherelationalstructureofthe world.Butthecorrespondencebetweenthemisnotanisomorphism–“themap isnottheterritory”,asKorzybskisaid(seeKorzybski1933,p.58)–butprobably notevenahomomorphism.Forexample,thestructureoftheworldmaybean asymmetricgraph,whilethestructureofthesystemissymmetric.However,we requirethesystemtorepresent,withoutnoticeabledistortion,themainfeatures oftheworldofwhichitissupposedtobeanimage.
Thefirsttwopostulateshaveamethodologicalcharacter.Thelatterisclearly ontological.
Thislatterisprobablythemostproblematic.Ithasbeenwidelydiscussed, sincethefamousarticlebytheAmericanmathematicianandphilosopherRandal RoyDipert(1997),ofwhichwewouldliketosayafewwordshere.
ThefirstpartofDipert’sarticle(“theshortcomingsoflogicandlogical metaphysics”)isoneofthebestarguedrefutationsofanalyticalphilosophy thatcanbefound.Thesecondpartarguesforawell-tempered“relationalism”. Butitisespeciallythethirdpart(“structure,asymmetricgraphsandAristotle refuted”)andthefourthpart(“theworldasasymmetricgraph”)whicharethe mostinterestingforourpurpose 7 .
5Thisprojectmustnotbeconsideredasanutopyoranold-fashioneddream.FromWhitehead toRescher,theideaofcognitivesystematizationhascontinued,as,moreover,itpersistsinour everydaylifewhenwetrytounderstandthisoneinallitsaspects.
6Letusjustsayhere,tofixideas,thatitisasetofverticesconnectedbyarcsoredges, dependingonwhetherthegraphisdirectedornot.
7Thereadercanfindthecompletedefinitionofsymmetricandasymmetricgraphsin section1.8.
impossible,isperfectlyaboutanobjectjustwhenitsinternal graphicalstructureisthesameastheobject’s,andwhentheobject occupiesalocationinthesystemofallobjectsthatislikethe thought’slocationinthesystemof(thatmind’s)thought(Dipert1997, p.357).
Wemightthinkthenthat,insuchatheory,thereisnoroomforminds, consciousnessandothermentalphenomena,unless,precisely,everythingis mental,whichfinally,Dipert,followinginthisLeibniz,ifnotSpinoza,does notentirelyexclude,becausetheverticesofthegraphcould,afterall,onlybe “feelings”.
Thisremarkablearticlesurelyunleashedthehostilityofthephilosopherswho preferAristotelianmattertoformalstructuresandthesentimentalillusionof humanwarmthtotheobjectiverelationshipsdepictedinthiscoldgraph.Butthe strongestobjection,developedinashortarticlebyOderberg(2011,pp.6–9), wasbasedonamathematicalargumentthatShackel(2011,p.11)verywell summarizedasfollows:
1)supposetheworldisagraph;
2)ifitisagraph,itisanasymmetricgraph;
3)anyasymmetricgraphcanbeturnedintosymmetricgraphbytheremoval ofedges;
4)thelossoflessthanalltheedgesoftheworld(specifically,thelossofall thosenotpartofsomesymmetricsubgraph,alossconsequentoncertainnodes goingoutofexistence)metaphysicallyentailsthenon-existenceoftheentire world(1,2,3);
5)Therefore,theworldisnotagraph(1,4,reductioadabsurdum).
Infact,thismathematicalargumentisquitespecious:whyimaginethatthe asymmetricgraphoftheworldcouldbecomesymmetrical?AsShackelwrites, “it’simpossiblefortheworldgraphtobeotherthanitisandhenceanychangeat allentailsnon-existence”(Shackel2011,p.13),soOderberg’sobjectionisnota validargument:“itisimpossibletogetridofthenodesandedgesofaworldand theargumenttotheabsurdinexistenceisblocked”(Shackel2011,p.14).
ThefactremainsthatthenecessityofDipert’sstructuremayseem troublesome.However,reducingthepropertiestosimplepotentials,asBirdcould havedoneinanothermodel(seeBird2007),wouldnotremovetheproblem accordingtoOderberg.Becauseifthereisonlypotentialandnoactualization,
thennothingcanreallymanifestitself,ashealsoexplainsinanotherarticle (seeOderberg2012).But,asShackelshows,therearepotencygraphsfor whichadiffusionprocessgraduallyupdatessome(orall)ofthepotentials.The phenomenonisa“snowball”andwecandemonstratethat,inthistypeofgraph (snowflakegraphs),wecanalwayschooseanodefromwhichallthepotentials canbeactualized.
Inconclusion,Dipert’sgraphisperfectlyplausibleandthispossiblereduction oftheworldtoagraphmakesitpossible,aswell,tosomehowontologically basetheattemptsofphilosophersovertime.Ifitisrejected,however,itcannotbe deniedthattheseattemptscanbemethodologicallydescribedusingtheformalism ofgraphtheory.Sowedonotabsolutelyneedtheassumption P3 tojustifywhat follows.
However,ifweaccepttoconsidertheworldasagraph,then,iftheworldis finite,thereisagoodchancethatitsgraphisasymmetric.Indeed,theproportion ofgraphson n verticeswithnon-trivialautomorphismstendstozerowhen n grows,whichinformallymeansthatalmostallfinitegraphsareasymmetric. Incontrast,almostallinfinitegraphsaresymmetricand,morespecifically, countableinfiniterandomgraphsintheErdös–Rényimodelare,withprobability 1,isomorphictothehighlysymmetricRadograph(seeErdösandSzekeres1935).
Inthistext,usingthelanguageofgraphs,ordersandsometimesinfinites,we trytorepresentmathematicallythisentitythatJasperscalled“theencompassing”, andwhichhethoughtitwas,bydefinition,impossibletoobjectify.
Infact,mathematicshasoftentaughtusthatitispossibletoobtaingood characterizationsofanobjectwithouthavingtobringitinanyexteriority.As Gaussshowedinthecaseofsurfaces(theoremaegregium),itisveryremarkable thatthecurvatureofageometricalobjectcanbedescribedintrinsically,i.e. withoutanyreferencetoa“spaceofembedding”inwhichtheconsideredobject wouldbelocated.Forexample,thefactthatanordinarysphereisasurfacewith constantpositivecurvatureiscompletelyindependentofthefactthatweusually seethissphereasbeingimmersedinourthree-dimensionalEuclideanspace. Thecurvatureofthisspherecouldverywellbemeasuredbytwo-dimensional intelligentbeingslivingonthesphere(kindsof“two-dimensionalants”),from measurementsoflengthsandanglescarriedoutonthesphere.
Wemustimaginethatweare,withrespecttotheencompassing,inthesame situationastwo-dimensionalantswithrespecttothesphere.Inprinciple,nothing preventsusfrombeingabletodescribeitifwehavethenecessaryinformation.
Thischapterisanintroductiontooursubject.Afterafewwordsaboutthe historyofgraphs,weexplainwhatwenowcalla“graph”inmathematicsand introducethemaindefinitionsusefultounderstandwhatfollows.Then,westudy someparticularclassesofgraphsthatwecanmeetinphilosophyandgivesome examplesofthem.
1.1.Graphtheory:abriefhistory
Theoriginofgraphdrawingsisnotwellknown,butwehavealreadyfound someoftheminancientChina.
Asintheterrestrialorder,wheretheancientChineseusedcryptographyto restrictthetransmissionofcertainmessagestotheinitiates,communication withtheheavenlyforcesrequiredinChinaaformadaptedtotheaddressee, andconverselythemessageofcelestialforcescouldnottakeovertheform ofaterrestrialmessage.Thuscamethe fu,thesegraphs,mostoftenDaoist, whichallowedfollowerstoenterintocommunicationwithspiritsandspiritsto communicatewiththeearthlyworld(seeFigure1.1).
Inthisregionoftheworld,somegraphicconstructionsmayalsobefoundin decorativemotivesofarchitectureorjails(seeFigure1.2)whichfollowcertain logics(HeandSchnabel,2018).
1Foracolorversionofallthefiguresinthischapter,seehttp://www.iste.co.uk/parrochia/ graphs.zip.
2Graphs,Orders,InfinitesandPhilosophy
Figure1.1. TaoistFusurmountedbyaconstellationof5stars(right);officialof fate(left)(BNF,Coins,MedalsandAntiques-CFA-135)(source:BNF)
Chinesedecorativegraphs(photo:Z.Guo)
Itseemsthatoneoftheearliestformsoftheminwesterncountriesare probablythatofMorrisandMillgames,asshowninFigure1.3,wherethenodes ofthegraphdrawingarethepositionsthatgamecounterscanoccupy,theedges indicatinghowgamecounterscanmovebetweennodes(seeKruskal1960,pp. 272–286).
Summer Palace (Beijing) - photo Z. Guo
Lingering Garden (Suzhou)
Forms of motives
Figure1.2.
Figure1.3. DepictionofMorrisgameboards
4Graphs,Orders,InfinitesandPhilosophy
paper,aswellastheonewrittenbyVandermondeontheknightprobleminchess, carriedonwiththedreamoftheso-called“analysissitus”initiatedbyLeibnizand theancestoroftopology.
Euler’sformularelatingthenumberofedges,verticesandfacesofaconvex polyhedronwasstudiedandgeneralizedbyCauchy(1813)andL’Huillier (1812-1813)andrepresentsthebeginningofthisnewbranchofmathematics: algebraicaltopology.
MorethanonecenturyafterEuler’spaperandwhileListingwasintroducing theconceptoftopology,Cayleywasledbyaninterestinparticularanalytical formsarisingfromdifferentialcalculustostudyaparticularclassofgraphs, thetrees(seeCayley1857).Thisstudywaspartlymotivatedbyimportant problemsintheoreticalchemistry.ThetechniquesCayleyusedmainlyconcerned theenumerationofgraphswithspecificproperties.Enumerativegraphtheory thenarosefromhisresultsandalsothefundamentalresultspublishedbetween 1935and1937.ThesewerethengeneralizedbyDeBruijnin1959.Thelinks establishedbyCayleybetweenhisresultsontreesandcontemporarystudiesof chemicalcomposition(seeCayley1875)influencedthedevelopmentofastandard terminologyingraphtheory.
Forexample,theterm“graph”wasintroducedbySylvesterinapaper publishedin1878in Nature,wherehedrawsananalogybetween“quantic invariants”and“co-variants”ofalgebraandmoleculardiagrams(seeSylvester 1878):
Everyinvariantandco-variantthusbecomesexpressiblebyagraph preciselyidenticalwithaKekuléandiagramorchemicograph.Igivea ruleforthegeometricalmultiplicationofgraphs,i.e.forconstructing agraphtotheproductofin-orco-variantswhoseseparategraphsare given.
Thefirsttextbookongraphtheorywas writtenbyDénesKönigandpublished in1936(seeKönig1990).AnotherbookbyFrankHarary,publishedin1969, was“consideredtheworldovertobethedefinitivetextbookonthesubject”(see Gardner1992,p.203)andenabledmathematicians,chemists,electricalengineers andsocialscientiststotalktoeachother.InFrance,ClaudeBerge(seeBerge 1970)hasalsopublishedinthe1970saveryimportantbookaboutgraphsand hypergraphs.
Letusnowentersomeparticularproblemsingraphtheory.
Oneofthemostfamousandstimulatingproblemsinthisfieldisthefourcolor problem:“Isittruethatanymapdrawnintheplanemayhaveitsregionscolored withfourcolors,insuchawaythatanytworegionshavingacommonborder havedifferentcolors?”Thisproblem wasfirstposedbyFrancisGuthriein1852 anditsfirstwrittenrecordisinaletterofDeMorganaddressedtoHamiltonthe sameyear.Manyincorrectproofshavebeenproposed,includingthosebyCayley, Kempeandothers.ThestudyandthegeneralizationofthisproblembyTait, Heawood,RamseyandHadwigerledtothestudyofthecoloringsofthegraphs embeddedonsurfaceswitharbitrarygenus.Tait’sreformulationgeneratedanew classofproblems,thefactorizationproblems,particularlystudiedbyPetersen andKönig.TheworksofRamseyoncolorationsandmorespeciallytheresults obtainedbyTuránin1941wereattheoriginofanotherbranchofgraphtheory, extremalgraphtheory.
Thefour-colorproblemremainedunsolvedformorethanacentury.In1969, HeinrichHeesch(seeHeandSchnabel2018)publishedamethodforsolving theproblemusingcomputers.Acomputer-aidedproofproducedin1976by KennethAppelandWolfgangHakenmakesthefundamentaluseofthenotion of“discharging”developedbyHeesch(seeAppelandHaken1977a,1977b). Theproofinvolvedcheckingthepropertiesof1,936configurationsbycomputer, andwasnotfullyacceptedatthetimeduetoitscomplexity.Asimplerproof consideringonly633configurationswasgiven20yearslaterbyRobertsonetal. (1997).
Theautonomousdevelopmentoftopologyfrom1860to1930fertilizedgraph theorybackthroughtheworksofJordan,KuratowskiandWhitney.Another importantfactorofcommondevelopmentofgraphtheoryandtopologycame fromtheuseofthetechniquesofmodernalgebra.Thefirstexampleofsuchause comesfromtheworkofthephysicistGustavKirchhoff,whopublishedin1845his Kirchhoff’scircuitlawsforcalculatingthevoltageandcurrentinelectriccircuits.
Theintroductionofprobabilisticmethods ingraphtheory,especiallyinthe studyofErdösandRényioftheasymptoticprobabilityofgraphconnectivity, gaverisetoyetanotherbranch,knownas randomgraphtheory,whichhasbeen afruitfulsourceofgraph-theoreticresults.
Theconsiderationofflowsandnetworksandthedevelopmentofthe Ford–Fulkersonmethodprovideanewapproachingraphtheory,combinatorics andthenalgorithmics.
Wewillfirstexplorethealreadylargeuniverseofgraphs.
1.2.Basicdefinitions
Letusbeginwithsomedefinitions.
D EFINITION 1.1.– Initsmodernmathematicalsense,agraphisanorderedpair G(V,E ) comprisingaset V ofverticesornodesorpointstogetherwithaset E ofedgesorarcsorlines,whichare2-elementsubsetsof V (i.e.anedgeis associatedwithtwovertices,andthatassociationtakestheformoftheunordered paircomprisingthosetwovertices).Thisisnotexactly,indeed,themostgeneral definition.Toavoidambiguity,wewillseelaterthatthistypeofgraphmaybe describedpreciselyasundirectedandsimple.Thischaracteristicsaresufficient forthemoment.
R EMARK 1.1.– Theverticesbelongingtoanedgearecalledtheendsorend verticesoftheedge.Avertexmayexistinagraphandnotbelongtoanedge.
R EMARK 1.2.– Thesets V and E areusuallytakentobefinite,andmanyofthe well-knownresultsarenottrue(orareratherdifferent)forinfinitegraphsbecause manyoftheargumentsfailintheinfinitecase.
D EFINITION 1.2.– Theorderofagraphis |V |,itsnumberofvertices.Thesize ofagraphis |E |,itsnumberofedges.Thedegreeorvalencyofavertexisthe numberofedgesthatconnecttoit,whereanedgethatconnectsavertextoitself (aloop)iscounted.
Foranedge {x,y },graphtheoristsusuallyusethesomewhatshorternotation xy
1.3.Differenttypesofgraphs
Aswehaveseen,manygraphshavebeendiscoveredthroughoutthehistory ofthediscipline,sothattoday,evenrestrictingustointerestingorwell-known graphs,whicharealreadyverynumerous,wearedealingwithanextremelylarge set,infactaverybushyforest,nottosayarealjungle.
Exploringthiscollectionsupposesthatwehaveidentifiedsomedistinctive featuresthatcanguideusandallowustodefineclassificationcriteria.
Thereare,infact,varioustypesofgraphsdependinguponthenumberof vertices,thenumberofedges,theirinterconnectivityandtheiroverallstructure. Wewilldiscussonlyacertainfewimportanttypesofgraphsinthischapter.
8Graphs,Orders,InfinitesandPhilosophy
D EFINITION 1.10(C YCLEGRAPH ).– Asimplegraphwith n vertices(n ≥ 3)and n edgesiscalledacyclegraphifallitsedgesformacycleoflength n
R EMARK 1.4.– Ifthedegreeofeachvertexinthegraphistwo,thenitiscalleda cyclegraph.
Philosophicalexample: AsinthesystemofHegel,thesequenceofthemain conceptsofthephilosophyofEricWeilformsacyclegraph.
D EFINITION 1.11(W HEELGRAPH ).– Awheelgraphisobtainedfromacycle graph Cn 1 byaddinganewvertex.ThatnewvertexiscalledaHubwhichis connectedtoalltheverticesof Cn .
Philosophicalexample: InLeibnizianphilosophy,Godisa Hub becauseHeis connectedtoallthemonads.Conversely,eachmonadisconnectedtoallaspects oftheworld.So,theLeibniziansystemis,fromagraphicpointofview,awheel ofwheels.
D EFINITION 1.12(C YCLICGRAPH ).– Agraphwithatleastonecycleiscalleda cyclicgraph.
Philosophicalexample: Therearealotofcyclesintheclassicphilosophical systemsofLeibnizorHegel.
D EFINITION 1.13(ACYCLICGRAPH ).– Agraphwithnocyclesiscalledan acyclicgraph.
Philosophicalexample: InDescartes’philosophy,the“longchainsofreasons” areexamplesofacyclicgraphs.ButDescartes’systemitselfisnotacyclic(see Beyssade1979;Parrochia1993b,p.183,2012,pp.114–115).
D EFINITION 1.14(B IPARTITEGRAPH ).– Asimplegraph G =(V,E ) witha vertexpartition V = {V1 ,V2 } iscalledabipartitegraphifeveryedgeof E joins avertexin V1 toavertexin V2
Philosophicalexample: AccordingtoGillesDeleuze(seeDeleuze2003, pp.42–43),inthewholegraphofSpinoza’s Ethics,wecanmakeapartition intotwobigclassesofverticesassociatedwithstatements:ontheonehand,the propositions,corollariesandproofs,whichformthedeductivepartofthebook; ontheotherhand,thescholia,adiscontinuousbrokenchainwhichdonotbelong tothededuction.However,scholiaareconnectedtopropositionsandcorollaries (ortheirproofs)bysomeedges.Allthatdonotconstitutereallyabipartitegraph.
