The complete graph maximizes the variance of the antivoter model

In the antivoter model on a finite graph each vertex, at rate one, adopts the opinion opposite to that of a uniformly chosen neighbour. Let $S$ be the sum of the $\pm1$ opinions in stationarity. Aldous and Fill asked whether, among regular graphs on $n$ vertices, $\operatorname{Var} S$ is largest for the complete graph, where it equals $n(n-2)/(2n-3)$. We prove that it is: for every connected non-bipartite regular graph on $n\ge3$ vertices and every stationary distribution, $\operatorname{Var} S\le n(n-2)/(2n-3)$, with equality only for $K_n$. The proof writes the pair correlations as the solution of a Sylvester equation and reduces the question to a lower bound on the diagonal of $(I+P)^{-1}$, where $P$ is the transition matrix of simple random walk. For vertex-transitive graphs the same argument gives $\operatorname{Var} S=n^2/\bigl(2\sum_i(1+\lambda_i)^{-1}\bigr)$, the sum running over the eigenvalues of $P$.