Among all graphs with $n$ vertices and edge-connectivity exactly $c$ (so the size of the minimum edge cut is $c$), there is a well-known result of Lomonsov and Poleskkii that the cycle graph, which consists of $n$ vertices arranged in a cycle, and $c/2$ parallel edges between adjacent vertices, has the fewest MINIMAL cuts (i.e. cuts of weight exactly c)
What happens if $c$ is odd? Strangely, all the references I can find to this result omit this case.
EDIT: I left out the condition that the graph has the fewest MINIMAL cuts. Sorry.