图若干矩阵表示的研究硕士论文

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1、分类号：O157.5密级：学校代码：10414学号：2009010851硕士研究生学位论文图的若干矩阵表示的研究Researchonsomematrixrepresentationsofgraphs苏莉院所：数学与信息科学学院导师姓名：赵宪钟学科专业：应用数学研究方向：代数图论二○一二年四月,,,DZDZ

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3、ractIn1957CollatzandSinogowitz’screativepaper[13]laidthegroundworkforspectralgraphtheory.However,theideaofspectralmethodperhapshadbeenfoundintheH¨uckel’swork[29]in1931,wherethespectrumofthecorrespondinggraphbearsasimplealgebraicrelationtothesetofpossibleπ-electronenergy

4、levels.EvenearlierworkontheaspectmaybetracedbacktoKirchhoff’swell-knownMatrix-treetheorem,whichwasobtainedwhenhestudiedtheelectronicnetworksbymeansofLaplacianspectra.Manyscholarsingraphtheory,combinatorics,theoreticalchemistryandphysicsenteredintoandthencontributedmuchto

5、thedevelopmentofthefield,whichmakespectralgraphtheorytobecomesodynamicandfascinatingaround30years.Spectralgraphtheoryhastraditionallyusedthespectraofspecificmatricesassociatedwiththegraph,suchastheadjacencymatrix,theLaplacianmatrix,orthedistancematrix,toprovideinformation

6、aboutthegraph.Ithasalonghistoryandisoneofthemostdynamicsubjectsingraphtheorywithnumerousapplicationsinotherfields,includingdatacommunicationproblem,theoreticalcomputerscience,distributedcontrolsystemsandchemistryscience.ThethesismainlyinvestigatethespectraofLaplacianofdi

7、graphs,signlessLaplaciananddistancematrixofundirectedgraphs.Itconsistsoffourchaptersandfocusesonthefollowingproblems:Inchapter1,weshallintroducethebasicconceptsandresults,andstatealsotherelatedresearchbackground.Inchapter2,weinvestigatetheessentialcyclicityofsomedigraph

8、s,includingdigraphsobtainedfromwheel,digraphswithpathstructureandsomedigraphswithHamiltoniancycle,solveouttheL