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 $\ell^2(V)$. Each vector $\xi \in \ell^2(V)$ defines a scalar-valued spectral measure $\mu_\xi$ on the spectrum of $A$, and in particular each vertex $v \in V$ defines a characteristic function which is a vector $\delta_v \in \ell^2(V)$, and a scalar-valued spectral measure $\mu_v$ on the spectrum of $A$. The purpose of this note is to call attention to dominant vertices, i.e., to vertices $v \in V$ such that the measure $\mu_v$ dominates $\mu_w$ for all other vertices $w \in V$ (hence dominates $\mu_\xi$ for all $\xi \in \ell^2(V)$). In some classes of graphs, for example in vertex transitive graphs, all vertices are dominant. It is also shown that there are graphs with only some dominant vertices, and graphs with no dominant vertices at all.