This paper studies the problem of finding the 1-median on a graph where vertex weights are uncertain and the uncertainty is characterized by given intervals. It is required to find a minmax regret solution, which minimizes the worst-case loss in the objective function. Averbakh and Berman had an O(mn²log n)-time algorithm for the problem on a general graph, and had an O(nlog² n)-time algorithm on a tree. In this paper, we improve these two bounds to O(mn² + n³log n) and O(nlog n), respectively.