A tight upper bound on the average order of dominating sets of a graph

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