  The availability of large datasets composed of graphs creates an
unprecedented need to invent novel tools in statistical learning for
graph-valued random variables. To characterize the average of a sample of
graphs, one can compute the sample Frechet mean and median graphs. In this
paper, we address the following foundational question: does a mean or median
graph inherit the structural properties of the graphs in the sample? An
important graph property is the edge density; we establish that edge density is
an hereditary property, which can be transmitted from a graph sample to its
sample Frechet mean or median graphs, irrespective of the method used to
estimate the mean or the median. Because of the prominence of the Frechet mean
in graph-valued machine learning, this novel theoretical result has some
significant practical consequences.
