Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A necessary and sufficient condition for a graph $G$, which satisfies
  the equality $\mu_{21}(G)=|V(G)|$   A necessary and sufficient condition is found for a graph $G$, which
satisfies the equality $\mu_{21}(G)=|V(G)|$.

8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th