Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Average (Edge-)Connectivity of Minimally $k$-(Edge-)Connected
  Graphs   Let $G$ be a graph of order $n$ and let $u,v$ be vertices of $G$. Let
$\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths
in $G$. Then the average connectivity $\overline{\kappa}(G)$ of $G$, is defined
as $ \overline{\kappa}(G)=\sum_{\{u,v\}\subseteq V(G)}
\kappa_G(u,v)/\tbinom{n}{2}. $ If $k \ge 1$ is an integer, then $G$ is
minimally $k$-connected if $\kappa(G)=k$ and $\kappa(G-e) < k$ for every edge
$e$ of $G$. We say that $G$ is an optimal minimally $k$-connected graph if $G$
has maximum average connectivity among all minimally $k$-connected graphs of
order $n$. Based on a recent structure result for minimally 2-connected graphs
we conjecture that, for every integer $k \ge3$, if $G$ is an optimal minimally
$k$-connected graph of order $n\geq 2k+1$, then $G$ is bipartite, with the set
of vertices of degree $k$ and the set of vertices of degree exceeding $k$ as
its partite sets. We show that if this conjecture is true, then
$\overline{\kappa}(G)< 9k/8$ for every minimally $k$-connected graph $G$. For
every $k \ge 3$, we describe an infinite family of minimally $k$-connected
graphs whose average connectivity is asymptotically $9k/8$. Analogous results
are established for the average edge-connectivity of minimally
$k$-edge-connected graphs.

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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: math-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: cond-mat