None. An edge must be straight.
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If a cylinder has two edges then a cone by that definition has one edge and one vertice.
One possibility is a cylinder whose bases do not stop in edges but flow gradually into the length of the cylinder.
It is the distance measured round its edge. multiply the radius (half the distance across it) by 3.14159
Any time we connect two edges we create a vertex. The only way a shape can have two edges and no vertices is if the edges are disconnected. For example, two parallel lines is two edges and no vertices. If you are thinking of polygons, we need at least 3 edges and they must be connected. So the answer to your question is really there is not shape that fits this description if we restrict ourselves to planar closed figures. The above is true for polyhedra. However, a cylinder meets the requirements if we consider the case where an edge is defined in such a way that a cylinder has two of them. The word "if" is there because sometimes, an edge is required to be straight. In that case a cylinder has none. However, many texts are not precise about how they define and edge and in this case we have no definition at all. So it is important to point out that the answer is a cylinder or nothing depending on your edge definition. If we use the definition of an edge as boundary between smooth surfaces making up an object, then a cylinder has two edges. There are similar issues with the term vertex and how it is used in general language, in toplogy and in graph theory. For this question, I think a good answer is a cylinder is the edges and vertex are appropriately defined. The link will help with more details for those interested.
A label on a bottle bought in a shop would show the volume. If there is no label, fill the bottle with water to the top, then empty the water into a measuring jar or cylinder. The volume is then read off the scale marked on the measuring jar or cylinder.