  As the counterpart of classical theorems on cycles of consecutive lengths due
to Bondy and Bollob\'as in spectral graph theory, Nikiforov proposed the
following open problem in 2008: 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 each integer $\ell\in[3,(C-\varepsilon)n]$. We
prove that $C\geq\frac{1}{4}$ by a novel method, improving the existing bounds.
Besides several novel ideas, our proof technique is partly inspirited by the
recent research on Ramsey numbers of star versus large even cycles due to
Allen, {\L}uczak, Polcyn and Zhang, and with aid of a powerful spectral
inequality. We also derive an Erd\H{o}s-Gallai-type edge number condition for
even cycles, which may be of independent interest.
