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1、數(shù)學(xué)雜志Vo1．34(2014)No．4MINIMUMDISTANCESPECTRALRADIUSoFGRAPHSWITHGIVENEDGECoNNECTIVITYLIXiao—xin1,2FANYi—zheng2WANGYi(．DepartmentMathematicsandComputerSciences，ChizhouUniversity，Chizhou247000，China)(2．SchoolofMathemntiealScie幾ces，AnhuiUniversity，Hefei230601，China)Abstract
2、：Inthispaperwestudytheextremalgraphswithminimumdistancespectralradiusamongallconnectedgraphsofordernandedgeconnectivityr．Byusingthecombinatorialmethod，wedeterminethatK(n一1，r)istheuniqueextremalgraph，whereg(n一1，r)isobtainedfromthecompletegraphKn一1byaddingavertexvtogeth
3、erwithedgesjoiningvtorverticesofKn一1．Alltheabovegeneralizetherelatedresultsoftheextremalgraphtheory．Keywords：graph；distancematrix；spectralradius；edgeconnectivity2010MRSubjectClassification：05C50Documentcode：AArticleID：0255—7797f2014)04-0671—081IntroductionLetGbeaconne
4、ctedsimplegraphwithvertexsetv(c)andedgesetE(G)．Thedistancebetweentwoverticesu，vofG，denotedbydis，isdefinedasthelengthoftheshortestpathbetweenandvinG．ThedistancematrixofG，denotedbyD(G)，isdefinedbyD(G)=(dis)u,vE(G)．SinceD(G)issymmetric，itseigenvaluesareallrea1．Inaddition
5、，asD(G)isnonnegativeandirreducible，byPerron—FrobeniustheoremIthespectralradiusp(G)ofD(G)(calledthedistancespectralradiusofG)，isexactlythelargesteigenvalueofD(C)withmultiplicityone；andthereexistsaunique(uptoamultiple)positiveeigenvectorcorrespondingtothiseigenvalue，usu
6、allyreferredtothePerronvectorofD(G)．Thedistancematrixisveryusefu1indiferentfieldsincludingthedesignofcommunica-tionnetworks[1]，graphembeddingtheory[2-4]aswellasmolecularstability[5，6]．Balabaneta1．[7]proposedtheuseofthedistancespectralradiusasamoleculardescriptor．Gutma
7、neta1．『8]usedthedistancespectralradiustoinfertheextentofbranchingandmode1boilingpointsofanalkane．Therefore，maximizingorminimizingthedistancespectralradiusoveragivenclassofgraphsisofgreatinterestandsignificance．Recently,themaximalortheReceiveddate：2013—04—13Accepteddat
8、e-"2013—06—13Foundationitern：SupportedbyNationalNaturalScienceFoundationofChina(11071002)；ProgramforNewCenturyExcellentTalen
