Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the second smallest and the largest normalized Laplacian eigenvalues
  of a graph   Let $G$ be a simple connected graph with order $n$. Let $\mathcal{L}(G)$ be
the normalized Laplacian matrix of $G$. Let $\lambda_{k}(G)$ be the $k$-th
smallest normalized Laplacian eigenvalue of $G$. Denote $\rho(A)$ the spectral
radius of the matrix $A$. In this paper, we study the behaviors of
$\lambda_{2}(G)$ and $\rho(\mathcal{L}(G))$ when the graph is perturbed by
three operations.

17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
37
The categories of this paper are: supr-con
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess