简介：GivengraphsG1andG2,wedefineagraphoperationonG1andG2,namelytheSSG-vertexjoinofG1andG2,denotedbyG1★G2.LetS(G)bethesubdivisiongraphofG.TheSSG-vertexjoinG1★G2.isthegraphobtainedfromS(G1)andS(G2)byjoiningeachvertexofG1witheachvertexof62-Inthispaper,whenGi(i=1,2)isaregulargraph,wedeterminethenormalizedLaplacianspectrumofG1★G2.Asapplications,weconstructmanypairsofnormalizedLaplaciancospectralgraphs,thenormalizedLaplacianenergy,andthedegreeKirchhoffindexofG1★G2.

简介：LetX,X1,X2,···beasequenceofnondegeneratei.i.d.randomvariableswithzeromeans,whichisinthedomainofattractionofthenormallaw.Let{ani,1≤i≤n,n≥1}beanarrayofrealnumberswithsomesuitableconditions.Inthispaper,weshowthatacentrallimittheoremforself-normalizedweightedsumsholds.WealsodeduceaversionofASCLTforself-normalizedweightedsums.