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1、LinearAlgebraanditsApplications506(2016)579–587ContentslistsavailableatScienceDirectLinearAlgebraanditsApplicationswww.elsevier.com/locate/laaLowerboundsofdistanceLaplacianspectralradiiofn-vertexgraphsintermsofmatchingnumberFengleiTian,DeinWong,1,XiaobinMaDepartmentofMathematics,ChinaUniversityofM
2、iningandTechnology,Xuzhou221116,PRChinaarticleinfoabstractArticlehistory:Received2March2016Accepted15June2016Availableonline23June2016SubmittedbyR.BrualdiMSC:05C5015A18Keywords:DistanceLaplacianmatrixSpectralradiusMatchingnumberRecently,Niuetal.(2015)[10]determinedtheextremalgraphswithminimumdist
3、anceLaplacianspectralradiusamongn-vertexbipartitegraphswithgivenmatchingnum-ber.However,amorenaturalproblemisleftopen:Amongalln-vertexgraphswithagivenmatchingnumberm,howaboutthelowerboundoftheirdistanceLaplacianspectralradiiandwhichgraphsminimizethedistanceLaplacianspectralradii?Inthisarticle,weso
4、lvethisproblemcompletely.DenotebyKmKn−mthegraphobtainedfromacompletegraphKmandn−misolatedverticesbyaddingm(n−m)edgesjoiningeachisolatedvertextoallverticesofKm.LetGbeaconnectedgraphofordernandmatchingnumberm,ρL(G)thedistanceLaplacianspectralradiusofG.Inthispaper,weprovethatifm=n2,thenρL(G)≥n,witheq
5、ualityifandonlyifG=Kn;andif1≤m≤n2−1,thenρL(G)≥2n−m,withequalityifandonlyifGΓ,whereΓdenotesthesetofn-vertexgraphsconsistingofKmKn−mtogetherwithallpossiblegraphsobtainedfromKmKn−mbydeletingsomeedgesofKm.©2016ElsevierInc.Allrightsreserved.*Correspondingauthor.E-mailaddress:wongdein@163.com(D.Wong).1S
6、upportedby“theNationalNaturalScienceFoundationofChina(11571360)”.http://dx.doi.org/10.1016/j.laa.2016.06.0280024-3795/©2016ElsevierInc.Allrightsreserved.580F.Tianetal./LinearAlgebraanditsApplications506(2016)579–5871.IntroductionLetGbeanundirectedconnectedsimplegraph.ThedistanceLaplacianmatrixofG,
7、definedasL(G)=diag(Tr)−D(G),andthedistancesignlessLaplacianmatrix,definedasQ(G)=diag(Tr)+D(G),haveattractedmuchattentionsincetheywereproposedbyAouchicheandHansenin[1],whereD(G)denotesthedistancematrixofGanddiag(T
