Spectral measures and dominant vertices in graphs of bounded degree

Abstract

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 $\Sigma (A)$ of $A$. In particular, each vector $\xi \in {\ell }^2(V)$ defines a local spectral measure ${\mu }_{\xi }$ at $\xi$ on $\Sigma (A)$; therefore each vertex $v\in V$ defines a vector ${\delta }_v\in {\ell }^2(V)$ and the associated measure ${\mu }_v$ on $\Sigma (A)$. A vertex $v$ is dominant if, for all $w\in V$, the measure ${\mu }_w$ is absolutely continuous with respect to ${\mu }_v$ (it then follows that, for all $\xi \in {\ell }^2(V)$, the measure ${\mu }_{\xi }$ is absolutely continuous with respect to ${\mu }_v$). 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.