  We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2
\rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex
triangle-free graph, and we make some progress toward proving this. We give
equivalent conditions for cycle-maximal triangle-free graphs; show bounds on
the numbers of cycles in graphs depending on numbers of vertices and edges,
girth, and homomorphisms to small fixed graphs; and use the bounds to show that
among regular graphs, the conjecture holds. We also consider graphs that are
close to being regular, with the minimum and maximum degrees differing by at
most a positive integer $k$. For $k=1$, we show that any such counterexamples
have $n\le 91$ and are not homomorphic to $C_5$; and for any fixed $k$ there
exists a finite upper bound on the number of vertices in a counterexample.
Finally, we describe an algorithm for efficiently computing the matrix
permanent (a #P-complete problem in general) in a special case used by our
bounds.
