I received a review of one of my papers in which the reviewer made an objection:
"...from the equations e(G) = v(G) you derive that there is a unique simple cycle of G. This is false for non-simple graphs (with loops)."
Here e(G) and v(G) are the number of edges and vertices of the graph, respectively. G is assumed to be connected.
I think the definition of a simple cycle is a path that begins and ends at the same vertex and does not repeat any vertices or edges (and uniqueness is up to cyclic permutation, i.e., the starting point doesn't matter).
I can't think of any counterexample even when I allow multiple edges between vertices, or loops (an edge from a vertex to itself). In fact, I feel like this should be easy to prove. Am I missing something?