论文标题
界图的光谱测量和主导顶点
Spectral measures and dominant vertices in graphs of bounded degree
论文作者
论文摘要
图形$ g =（v，e）的界限$具有邻接运算符〜$ a $，该$ a $作用于Hilbert Space $ \ ell^2（v）$。 $ a $ $ a $的频谱$σ（a）$σ有不同类型的措施。特别是，\ ell^2（v）$ in \ ell^2（v）$ in $σ$ in $σ（a）$上的每个向量$ξ\ in \ ell^2（v）$;因此，每个顶点$ v \ in v $ in \ ell^2（v）$定义了vector $Δ_v\ un $σ（a）$的关联度量$μ_v$。对于$μ_v$，对于所有$ w \ in v $中的所有$ w \，对于$μ_v$，$μ_w$绝对是连续的（然后遵循所有$ξ\ in \ ell^2（v）$，该量$μ_$ __en配对绝对连续$μ_v$）。本文的主要目的是表明所有可能性都发生了：在某些图中，例如在顶点传播图中，所有顶点都是主导的；在其他图中，只有一些顶点是主导的。并且有没有主要顶点的图形。
A graph $G = (V, E)$ of bounded degree has an adjacency operator~$A$ which acts on the Hilbert space $\ell^2(V)$. There are different kinds of measures of interest on the spectrum $Σ(A)$ of $A$. In particular, each vector $ξ\in \ell^2(V)$ defines a local spectral measure $μ_ξ$ at $ξ$ on $Σ(A)$; therefore each vertex $v \in V$ defines a vector $δ_v \in \ell^2(V)$ and the associated measure $μ_v$ on $Σ(A)$. A vertex $v$ is dominant if, for all $w \in V$, the measure $μ_w$ is absolutely continuous with respect to $μ_v$ (it then follows that, for all $ξ\in \ell^2(V)$, the measure $μ_ξ$ is absolutely continuous with respect to $μ_v$). The main object of this paper is to show that all possibilities occur: in some graphs, for example in vertex-transitive graphs, all vertices are dominant; in other graphs, only some vertices are dominant; and there are graphs without dominant vertices at all.
