All graphs discussed are finite and simple. The cycle sequence of a graph $G$, denoted $C(G)$, is the nondecreasing sequence of the lengths of all of the cycles in $G$, where cycles are distinguished by the vertices they contain, not by the edges they contain.
For example, $C(K_{3,2})=4,4,4$ and $C(K_4)=3,3,3,3,4$.
Two graphs are isoparic if they have the same number of vertices and the same number of edges.
Main question: If $G$ and $H$ are 2-connected nonisoparic graphs, can $C(G)=C(H)$?
The 2-connected condition is so we can't just make a bunch of edge-disjoint cycles that share a vertex. The nonisoparic condition is so we can ignore situations like the following:
These graphs are not isomorphic but are isoparic. Both graphs have the cycle sequence $3,3,4,5,5,6$ and can be viewed as just a square surrounded by two triangles. Perhaps there's a better way to ignore this trick besides the nonisoparic condition.
I'm interested more generally in finding out exactly what the cycle sequence can tell us. When is a cycle sequence realizable by a 2-connected graph? Is such a realization ever unique? I've looked at a couple dozen graphs on fewer than seven vertices and the only duplicate cycle sequences have been for the graphs shown above.
Thank you.