  Woodall proved that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is
an integer, if $e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$ then $G$ contains a
$C_{\ell}$ for each $\ell\in [3,n-k]$. In this article, we prove a stability
result of this theorem. As a byproduct, we give complete solutions to two
problems in \cite{GN19}. Our second part is devoted to an open problem by
Nikiforov: what is the maximum $C$ such that for all positive $\varepsilon<C$
and sufficiently large $n$, every graph $G$ of order $n$ with spectral radius
$\rho(G)>\sqrt{\lfloor\frac{n^2}{4}\rfloor}$ contains a cycle of length $\ell$
for every $\ell\leq (C-\varepsilon)n$. We prove that $C\geq\frac{1}{4}$ by a
method different from previous ones, improving the existing bounds. We also
derive an Erd\H{o}s-Gallai type edge number condition for even cycles, which
may be of independent interest.
