Optimal accuracy for linear sets of equations with the graph Laplacian

We show that certain Graph Laplacian linear sets of equations exhibit optimal accuracy, guaranteeing that the relative error is no larger than the norm of the relative residual and that optimality occurs for carefully chosen right-hand sides. Such sets of equations arise in PageRank and Markov chain theory. We establish new relationships among the PageRank teleportation parameter, the Markov chain discount, and approximations to linear sets of equations. The set of optimally accurate systems can be separated into two groups for an undirected graph -- those that achieve optimality asymptotically with the graph size and those that do not -- determined by the angle between the right-hand side of the linear system and the vector of all ones. We provide supporting numerical experiments.