  This is the third in a series of articles devoted to showing that a typical
covering map of large degree to a fixed, regular graph has its new adjacency
eigenvalues within the bound conjectured by Alon for random regular graphs.
  In this paper we consider random graphs that are random covering graphs of
large degree $n$ of a fixed base graph. We prove the existence of asympototic
expansion in $1/n$ for the expected value of the number of strictly
non-backtracking closed walks of length $k$ times the indicator function that
the graph is free of certain {\em tangles}; moreover, we prove that the
coefficients of these expansions are "nice functions" of $k$, namely
approximately equal to a sum of polynomials in $k$ times exponential functions
of $k$.
  Our results use the methods of Friedman used to resolve Alon's original
conjecture, combined with the results of Article~II in this series of articles.
One simplification in this article over the previous methods of Friedman is
that the "regularlized traces" used in this article, which we call {\em
certified traces}, are far easier to define and work with than the previously
utilized {\em selective traces}.
