On the maximum number of cycles in a Hamiltonian graph

Let image denote the maximum number of cycles in a Hamiltonian graph of order n and size image. We prove that image and image for image. Furthermore, we determine image and the structure of the extremal graphs for image exactly. Our results give partial answers to a problem raised by Shi [The number of cycles in a hamilton graph, Discrete Math. 133 (1994) 249–257].