For a nontrivial connected graph of order and a cyclic ordering of , let , where is the distance between and for . The Hamiltonian number and the upper Hamiltonian number of are defined as:
, where the minimum is taken over all cyclic orderings of .
, where the maximum is taken over all cyclic orderings of .
All connected graphs with and have been characterized in [6, 13]. In this note, we first present a new and significantly improved proof of the characterization of all graphs whose Hamiltonian and upper Hamiltonian numbers differ by 1. We then determine all pairs of integers that can be realized as the order and upper Hamiltonian number of some tree.