某些附有自由积群的无限次元Cayley图的谱分解

Spectra for Some Infinite Cayley Graphs Associated with Free Product Groups

主要研究附有两个巡回群的自由积G=Zn%Zm(n,m≥2)的Cayley图的离散邻接作用素A的谱分解。Gr,(Zn%Zm)是由un=vm=1这样两个Unitary作用素u,v生成的C,-环,它的构造是已知的。若令n≥3时,T1=u+u,T2=v+v,;n=m=2时,T1=u,T2=v,那么A=T1+T2,T1,T2在(Cr,(G),τG)上是自由独立的,利用T1,T2各自的分布,使用自由合成积,就可以求出T1+T2的分布。

Presents the spectra A for some infinite Cayley graphs associated with free product groups G=Zn^＊Zm（n,m≥2） .Gr^＊（Zn^＊Zm）is the C-ring coming from two unitary operators u and v, u^n=v^m=1, which is given. If n≥3 T1=u＋u^＊,T2=v＋v^＊,and if n=m=2 ,T1=u,T2=v As is verified, A=T1＋T2＋T1＋T2are two free random variables in （C1^＊（G）,τG） The distribution for T1＋T2 is inferred by the uses of free convolution and the respective distribution of T1,T2.