Spectral measures and dominant vertices in graphs of bounded degree

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.