A tight upper bound on the average order of dominating sets of a graph - Beaton, Iain; Cameron, Ben (Journal of graph theory, 2024)
In this paper we study the average order of dominating sets in a graph, avd ( G ) $\,\text{avd}\,(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs G $G$ of order n $n$ without isolated vertices, avd ( G ) ≤ 2 n / 3 $\,\text{avd}\,(G)\le 2n/3$. Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have avd ( G ) = 2 n / 3 $\,\text{avd}\,(G)=2n/3$. We also use our bounds to prove an average version of Vizing's conjecture.