Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Eigenvalues and cycles of consecutive lengths   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.

16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
1
The categories of this paper are: cs
36
The categories of this paper are: cmp-lg
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph