Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Note on the minimal size of a graph with generalized connectivity
  kappa_3= 2   The concept of generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$
was introduced by Chartrand et al. in recent years. In our early paper,
extremal theory for this graph parameter was started. We determined the minimal
number of edges of a graph of order $n$ with $\kappa_{3}= 2$, i.e., for a graph
$G$ of order $n$ and size $e(G)$ with $\kappa_{3}(G)= 2$, we proved that
$e(G)\geq (6/5)n$, and the lower bound is sharp by constructing a class of
graphs, only for $n\equiv 0 \ (mod \ 5)$ and $n\neq 10$. In this paper, we
improve the lower bound to $\lceil(6/5)n\rceil$. Moreover, we show that for all
$n\geq 4$ but $n= 9, 10$, there always exists a graph of order $n$ with
$\kappa_{3}= 2$ whose size attains the lower bound $\lceil(6/5)n\rceil$.
Whereas for $n= 9, 10$ we give examples to show that $\lceil(6/5)n\rceil+1$ is
the best possible lower bound. This gives a clear picture on the minimal size
of a graph of order $n$ with generalized connectivity $\kappa_{3}= 2$.

25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: mtrl-th
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The categories of this paper are: eess
4
The categories of this paper are: cond-mat