Best constants for the Hardy–Littlewood maximal operator on finite graphs

We study the behavior of averages for functions defined on finite graphs G  , in terms of the Hardy–Littlewood maximal operator MGMG. We explore the relationship between the geometry of a graph and its maximal operator and prove that MGMG completely determines G (even though embedding properties for the graphs do not imply pointwise inequalities for the maximal operators). Optimal bounds for the p-(quasi)norm of a general graph G   in the range 0