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.