Triangle-free graphs with the maximum number of cycles

It is shown that for n≥141n≥141, among all triangle-free graphs on nn vertices, the balanced complete bipartite graph K⌈n/2⌉,⌊n/2⌋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⌋K⌈n/2⌉,⌊n/2⌋. Also, an upper bound for the number of Hamiltonian cycles in a triangle-free graph is given.