Before revision: In this note we prove the following result: the sample Fr\'echet mean of a set of sparse graphs is sparse.
After revision: 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 Fr\'echet mean. Because the sample mean should provide an interpretable summary of the graph sample, one would expect that the structural properties of the sample be transmitted to the Fr\'echet mean. In this paper, we address the following foundational question: does the sample Fr\'echet mean inherit the structural properties of the graphs in the sample? Specifically, we prove the following result: the sample Fr\'echet mean of a set of sparse graphs is sparse.