Given a set of elements, the median can be a useful concept to get a representative that captures the global information of the set. In the domain of structural pattern recognition, the median of a set of graphs has also been defined and some properties have been derived. In addition, the maximum common subgraph of a set of graphs is a well known concept that has various applications in pattern recognition. The computation of both the median and the maximum common subgraph are highly complex tasks. Therefore, for practical reasons, some strategies are used to reduce the search space and obtain approximate solutions for the median graph. The bounds on the sum of distances of the median graph to all the graphs in the set turns out to be useful in the definition of such strategies. In this paper, we reduce the upper bound of the sum of distances of the median graph and we relate it to the maximum common subgraph.