A median of a k-tuble ≠ = (x1,…,xk) of vertices of a finite connected graph G is a vertex x for which Σki=1d(x,xi) is minimum, where d is the geodesic metric on G. The function M with domain the set of all k-tuples with k > 0 and defined by M(≠)={x| x is a median of ≠} is called the median function on G. In this paper a new characterization of the median function is given for G a median graph. This is used to give a characterization of the median function on median semilattices.
