地图计数通论
出版社：科学出版社
年代：2009
定价：108.0
书籍简介:
本书除改进和简化专著《Enumerative Theory of Maps》中的结果外，主要增添如下的新结果：1，在第1章，提供此论变换计数函数的3个新的基本定理。2，在第2章，用无穷矩阵幂，导出平面树的最终结果。3，在第3章，讨论小亏格曲面上的三角化。4，在第4章，给出3-正则c-网的无和显式。5，在第5章，导出曲面上Euler地图的计数方程，提供递推公式。6，在第6章，导出曲面上不可分离地图的计数方程，提供递推公式。7，在第7章，导出曲面上无环地图的计数方程，提供递推公式。8，在第8章，导出曲面上一般地图的计数方程，提供递推公式。9，9--10，在9-10章，导出曲面上的双不可分离地图的色和与梵和所满足的方程。在第11章，讨论上圈定向地图，在曲面上以亏格为参数的计数显式。12--13，在12-13章，讨论联树以及由此导出地图与图的嵌入之间的转换。在14-15章，提供从带根到无根，和从平面到非平面的一些原理。另外，书末还附有用联树模型表示的小地图在曲面上的附表。
书籍目录:
Preface
Chapter1Preliminaries
§1.1Maps
§1.2Polynomialsonmaps
§1.3Enufunctions
§1.4Polysumfunctions
§1.5TheLagrangianinversion
§1.6Theshadowfunctional
§1.7Asymptoticestimation
§1.8Notes
Chapter2OuterplanarMaps
§2.1Planetrees
§2.2Wintersweets
§2.3Unicyclicmaps
§2.4Generalouterplanarmaps
§2.5Notes
Chapter3Triangulations
§3.1Outerplanartriangulations
§3.2Planartriangulations
§3.3Triangulationsonthedisc
§3.4Triangulationsontheprojectiveplane
§3.5Triangulationsonthetorus
§3.6Notes
Chapter4Quadrangulations
§4.1Outerplanarquadrangulations
§4.2Outerplanarquadrangulationsonthedisc
§4.3Hamiltonianquadrangulationsonthesphere
§4.4Innerendlessplanarquadrangulations
§4.5Quadrangulationsontheprojectiveplane
§4.6QuadrangulationsontheKleinbottle
§4.7Notes
Chapter5EulerianMaps
§5.1PlanarEulerianmaps
§5.2Tutteformula
§5.3Eulerianplanartriangulations
§5.4RegularEulerianplanarmaps
§5.5Eulerianmapsonsurfaces
§5.6Notes
Chapter6NonseparableMaps
§6.1Outerplanarnonseparablemaps
§6.2Euleriannonseparablemaps
§6.3Planarnonseparablemaps
§6.4Nonseparablemapsonsurfaces
§6.5Bridgelessmapsonsurfaces
§6.6Notes
Chapter7SimpleMaps
§7.1Looplessmaps
§7.2Generalsimplemaps
§7.3Simplebipartitemaps
§7.4Looplessmapsonsurfaces
§7.5Notes
Chapter8GeneralMaps
§8.1Generalplanarmaps
§8.2Planarc-nets
§8.3Convexpolyhedra
§8.4Quadrangulationsviac-nets
§8.5Generalmapsonsurfaces
§8.6Notes
Chapter9ChrosumEquations
§9.1Treeequations
§9.2Outerplanarequations
§9.3Generalequations
§9.4Triangulationequations
§9.5Welldefinedness
§9.6Chrosumsonsurfaces
§9.7Notes
Chapter10PolysumEquations
§10.1Polysumsforbitrees
§10.2Outerplanarpolysums
§10.3Generalpolysums
§10.4Nonseparablepolysums
§10.5Polysumsonsurfaces
§10.6Notes
Chapter11MapsviaEmbeddings
§11.1Automorphismgroupofagraph
§11.2Embeddingsofagraph
§11.3Supermapsofagraph
§11.4Mapsfromembeddings
§11.5Notes
Chapter12LocallyOrientedMaps
§12.1PlanarHamiltonianmaps
§12.2Biboundaryinnerrootedmaps
§12.3Boundarymaps
§12.4Cubicboundarymaps
§12.5Notes
Chapter13GenusPolynomialsofGraphs
§13.1Jointtreemodel
§13.2Layerdivisions
§13.3Graphsfromsmaller
§13.4Pan-bouquets
§13.5Notes
Chapter14FromRootedtoUnrooted
§14.1Symmetricrelations
§14.2Anapplication
§14.3Symmetricprinciples
§14.4Generalexamples
§14.5Fromundergraphs
§14.6Notes
Chapter15FromPlanartoNonplanar
§15.1Treeswithboundary
§15.2Cuttingalongvertices
§15.3Cuttingalongfaces
§15.4Mapswithaplanebase
§15.5Vertexpartition
§15.6Notes
Chapter16ChromaticSolutions
§16.1Generalsolution
§16.2Cubictriangles
§16.3Invariants
§16.4Fourcolorsolutions
§16.5Notes
Chapter17StochasticBehaviors
§17.1Asymptoticsforouterplanarmaps
§17.2Theaverageontree-rootedmaps
§17.3Hamiltoniancircuitspermap
§17.4Theasymmetryonmaps
§17.5Asymptoticsviaequations
§17.6Notes
AppendixAtlasofSuperMapsforSmallGraphs
Ax.1BouquetsBm,4≥m≥1
Ax.2LinkbundlesLm,6≥m≥3
Ax.3CompletebipartitegraphsKm,n,4≥m,n≥3
Ax.4WheelsWn,5≥n≥4
Ax.5Triconnectedcubicgraphsofsizein[6,15]
Bibliography
SubjectIndex
AuthorIndex
内容摘要:
Thismonographisfortheoryanditsextensionsrelatedtocensusofmapsongeneralsurfacesestablishedonthebasisofwhathasbeendoneonthesurfaceofgenuszero.Inspiteofanumberofimprovedresultswithmapsonasurfaceofgenuszeroforsurfacesofgenusnotzero,twonewtheoreticallinesviaexploitingtherelationshipbetweensupermapsandembeddingsofagraphonsurfacesandviathejointtreetechniqueareinvestigatedforavarietyoftopicssuchasthoseinthedeterminationofhandleandcrosscappolynomialsofmaps,genusdistributionofembeddings,andothersrelated.Inparticular,anappendixservesastheexhaustivecountingsupermaps(rootedandnonrooted)includingthesepolynomialswithundergraphsofsmallsizeforthereadersdigests. SincethefirstmonographtitledEnumerativeTheoryofMapsappearedonthesubjectconsideredin1999,manyadvanceshavebeenmadebytheauthorhimselfandthosedirectedbyhimundersuchatheoreticalfoundation. Becauseofthatbookwithmuchattentiontomapsonsurfaceofgenuszero,thismonographisinprincipleconcernedwithmapsonsurfacesofgenusnotzero.Viamaintheoreticallines,thisbookisdividedintofourpartsexceptChapter1forpreliminaries. PartonecontainsChapters2through10.Thetheoryispresentedformapsongeneralsurfacesofgenusnotnecessarytobezero.Forthetheoryonasurfaceofgenuszeroiscomprehensivelyimprovedforinvestigatingmapsonallsurfacesofgeneranotzero. ParttwoconsistsofonlyChapter11.Relationshipsareestablishedforbuildingupabridgebetweensupermapsandembeddingsofagraphviatheirautomorphismgroups. PartthreeconsistsofChapters12and13.Ageneraltheoryforfindinggenusdistributionofgraphembeddings,handlepolynomialsandcrosscappolynomialsofsupermapsareconstructedonthebasisofthejointtreemethodwhichenablesustotransformaprobleminahighdimensionalspaceintoaproblemonapolygon. Allotherchapters,i.e.,Chapters14through17,aspartfourareconcernedwithseveralaspectsofmainextensionstodistinctdirections. Anappendixservesasatlasofsupermapsoftypicalgraphsofsmallsizeonsurfacesfortheconvenienceofreaderstochecktheirunderstanding.
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