Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A characterization of the edge connectivity of direct products of graphs   The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the
graph, denoted as $G\times H$, with vertex set $V(G\times H)=V(G)\times V(H)$,
where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$ if
$x_1x_2\in E(G)$ and $y_1y_2\in E(H)$. The edge connectivity of a graph $G$,
denoted as $\lambda(G)$, is the size of a minimum edge-cut in $G$. We introduce
a function $\psi$ and prove the following formula %for the edge-connectivity of
direct products $$\lambda (G\times H)=\min
{2\lambda(G)|E(H)|,2\lambda(H)|E(G)|,\delta(G\times H), \psi(G,H), \psi(H,G)}
.$$ We also describe the structure of every minimum edge-cut in $G\times H$.

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The categories of this paper are: eess
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The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
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The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
31
The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
36
The categories of this paper are: cmp-lg
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The categories of this paper are: econ