Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Open problem on $\sigma$-invariant   Let $G$ be a graph of order $n$ with $m$ edges. Also let $\mu_1\geq \mu_2\geq
\cdots\geq \mu_{n-1}\geq \mu_n=0$ be the Laplacian eigenvalues of graph $G$ and
let $\sigma=\sigma(G)$ $(1\leq \sigma\leq n)$ be the largest positive integer
such that $\mu_{\sigma}\geq \frac{2m}{n}$. In this paper, we prove that
$\mu_2(G)\geq \frac{2m}{n}$ for almost all graphs. Moreover, we characterize
the extremal graphs for any graphs. Finally, we provide the answer to Problem 3
in \cite{KMT}, that is, the characterization of all graphs with $\sigma=1$.

27
The categories of this paper are: comp-gas
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
7
The categories of this paper are: hep-th
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci