It is shown that for n≥141, among all triangle-free graphs on n vertices, the balanced complete bipartite graph K⌈n/2⌉,⌊n/2⌋ is the unique triangle-free graph with the maximum number of cycles. Using modified Bessel functions, tight estimates are given for the number of cycles in K⌈n/2⌉,⌊n/2⌋. Also, an upper bound for the number of Hamiltonian cycles in a triangle-free graph is given.
Arman, A., Gunderson, D. S., & Tsaturian, S. (2016). Triangle-free graphs with the maximum number of cycles. Discrete Mathematics, 339(2), 699–711. https://doi.org/10.1016/j.disc.2015.10.008