  We prove a moderate deviation principle for subgraph count statistics of
Erdos-Renyi random graphs. This is equivalent in showing a moderate deviation
principle for the trace of a power of a Bernoulli random matrix. It is done via
an estimation of the log-Laplace transform and the Gaertner-Ellis theorem. We
obtain upper bounds on the upper tail probabilities of the number of
occurrences of small subgraphs. The method of proof is used to show
supplemental moderate deviation principles for a class of symmetric statistics,
including non-degenerate U-statistics with independent or Markovian entries.
