Abstract
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.
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