There is a Shape section in the Cloth settings where you can use vertex groups to make parts of the mesh stiffer.
So, not really edges but vertices, as far as I know. But I’m not very experienced with cloth simulation.
I didn’t use stiffness. Instead I modeled the shape a bit different.
And if you convert the cloth simulation to a shape key you can then adjust the shape or combine it with other shape keys to get what you want.