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 $x_1{x}_2\in E(G)$ and $y_1{y}_2\in E(H)$. 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 $$\lambda (G\times H)=\min 2\lambda (G)\mathrm{\mid }E(H)\mathrm{\mid },2\lambda (H)\mathrm{\mid }E(G)\mathrm{\mid },\delta (G\times H),\psi (G,H),\psi (H,G)\mathrm{.}$$ We also describe the structure of every minimum edge-cut in $G\times H$.