Before revision:  We prove the result for the graph Hamming distance, and the spectral adjacency pseudometric, using very different arguments  .
After revision:  We prove the result for the graph Hamming distance, and the spectral adjacency pseudometric, using very different arguments . In fact, we prove a stronger result: the edge density of the sample Fr\'echet mean is bounded by the edge density of the graphs in the sample. This result guarantees that sparsity is an hereditary property, which can be transmitted from a graph sample to its sample Fr\'echet mean, irrespective of the method used to estimate the sample Fr\'echet mean .