How many minimum edges in a Cyclic graph with n vertices?

The term "cyclic graph" is not well-defined.

If you mean a graph that is not acyclic, then the answer is 3. That would be the union of a complete graph on 3 vertices and any number of isolated vertices.

If you mean a graph that is (isomorphic to) a cycle, then the answer is n.

If you are really asking the maximum number of edges, then that would be the triangle numbers such as n (n-1) /2.