Let $\Gamma$ be an undirected finite graph. Write $M_r$ for the number of non-backtracking closed walks of length $r$ in $\Gamma$, i.e., walks where a step is never
followed by its inverse. Let $A$ be the adjacency matrix of $A_r$, and $\lambda_1,...,\lambda_N$ its eigenvalues.

If $\Gamma$ is regular of degree $d$, then

$$M_{2 k} = (d-1)^k \sum_{n=1}^N \left(U_{2 k}\left(\frac{\lambda_n}{2 \sqrt{d-1}}\right)
- \frac{1}{d-1} U_{2 k-2}\left(\frac{\lambda_n}{2 \sqrt{d-1}}\right)
\right),$$
where $U_m$ is the $m$th Chebyshev polynomial of the second kind. For a proof, see, e.g., section 1.4 of the Davidoff-Sarnak-Valette monograph.


In particular, if many ($\geq \epsilon N$, say) of the eigenvalues $\lambda_i$ are large ($\geq \Lambda$, where $\Lambda > 5 \sqrt{d}$, say) then, for $k$ not too small, $M_{2 k}$ is large (in fact, roughly in the order of $\epsilon N\cdot \Lambda^{2 k}$).

Is an expression such as the above possible when $\Gamma$ is not regular? Can we at least obtain the conclusion that, if many of the eigenvalues are large ($\geq \Lambda$, where $\Lambda \ggg \sqrt{D}$, and $D$ is the maximal degree of the vertices of $\Gamma$) then $M_{2 k}$ must be large?

(Related, vaguer question: https://mathoverflow.net/questions/380902/closed-geodesics-and-eigenvalues-in-a-non-regular-graph?noredirect=1#comment967890_380902)