| Abstract: | 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 ▫$$lambda (G times H) = min {2lambda(G)|E(H)|, 2lambda(H)|E(G)|, delta(G times H), psi(G,H), psi(H,G)} .$$▫ We also describe the structure of every minimum edge-cut in ▫$G times H$▫. |
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