  In this paper we study the the average order of dominating sets in a graph,
$\operatorname{avd}(G)$. Like other average graph parameters, the extremal
graphs are of interest. Beaton and Brown (2021) conjectured that for all graphs
$G$ of order $n$ without isolated vertices, $\operatorname{avd}(G) \leq 2n/3$.
Recently, Erey (2021) proved the conjecture for forests without isolated
vertices. In this paper we prove the conjecture and classify which graphs have
$\operatorname{avd}(G) = 2n/3$. We also use our bounds to prove the average
version of Vizing's Conjecture.
