The minimum and maximum edges' number of a graph

Question

Question: If a graph $G$ has $30$ vertices and only one cycle, what is the minimum and maximum number of edges?
Can anyone explain that?

I think the minimum number will equal the number of vertices because we have a cycle.

Answer 1

check this https://en.m.wikipedia.org/wiki/Complete_graph#Properties Maximum number of edges in a simple graph?

Answer 2

Hints:

The minimum will be achieved by the graph consisting of the shortest possible cycle and a bunch of isolated vertices. (There's no condition on the graph being connected.)

Assuming we're working with simple graphs, the shortest cycle is the $3$-cycle. So $K_3 \cup 27K_1$ contains the fewest cycles in a graph with a single cycle, drawn below:

For the maximum: choose a graph that achieves the maximum and consider what happens when we delete an edge from the cycle. We get an acyclic graph (a forest). Which forest has the most edges?

Extending the hint, this graph has $30$ vertices and one cycle:

The challenging part is to prove that no more edges are possible in a single-cycle graph.