Let $A$ be the adjacency matrix of a (non-oriented) graph $\Gamma$. Then $\textrm{Tr} A^k$ equals both the sum $\sum_i \lambda_i^k$ of $k$th powers of eigenvalues of $A$, on the one hand, and the number of closed paths of length $k$ in $\Gamma$, on the other hand. If we have a good bound on the latter, then we have a good bound on the former. For $k$ even, a good bound on the former quantity implies that there aren't too many large eigenvalues $\lambda_i$.
What happens if we have an even better bound on closed trails of length $k$, that is, closed paths where no edges are repeated? Or what happens if we can give a very good bound on the number of closed paths that neither backtrack (i.e., an edge is never followed by its inverse) nor have long repeated sequences of edges (as in, an edge sequence $$(v_0,v_1), (v_1,v_2),\dotsc, (v_{l-1},v_l)$$ appearing at least twice, where $l$ is not that much smaller than $k$)? Can we then say something stronger about $A$?
Here, by "a very good bound", I mean a bound that is stronger ( = smaller) than would be possible for the number of closed paths of length $k$: there are always plenty of backtracking closed paths (at least $d^{k/2}$ starting from a vertex of degree $d$).
(I've asked related questions in the recent past: see Closed paths, traces and spectra and Ihara zeta function and closed paths and trails .)
