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.

simple cyclesin $G$---no repetitions of vertices and edges allowed in a cycle? $\endgroup$ – Joseph O'Rourke Jan 24 '15 at 1:442more comments