Torusenes are defined as closed toroidal networks where every vertex or atom is 3-valent, and they can represent pure carbon tori. Here we study the geometries of two classes: hexagonal torusenes containing purely polyhex networks and the second class, 5,6,7-ring torusenes which besides hexagons contain also an equal number of 5- and 7-membered rings. As sophisticated quantum-mechanical methods for geometry determination are time consuming for large carbon cages, and having in mind the huge number of their isomers, one is interested in methods which are simple to apply but which are still able to produce plausible geometries. One of them is offered by the adjacency matrix elgenvectors (AME) method, which was proposed in this journal [D. E. Manolopoulos and P. W. Fowler, J. Chem. Phys. 96, 7603 (1992)]. The application of the AME method to fullerenes is based on an appropriately chosen triplet of eigenvectors. A rational choice may be made on the basis of their nodal properties. No rules have been formulated up to now on how to apply the AME method to torusenes. In order to find such a rule a systematic study of nodal properties of torusenes is crucial, and such a study is the subject of this paper. Theoretical and computer experimental considerations presented here suggest that a triplet a(2),a(3),a(opt) fulfills the task where the a(opt) should be checked for among those eigenvectors which possess no radial nodal plane but have one axial cut. In the present paper these findings have been elaborated for 5,6,7-ring torusenes with up to 270 atoms, and computer experiments have shown that similar findings hold for purely polyhex torusenes with up to 224 carbon atoms as well. In order to understand better these nodal properties, a quantum-mechanical study of free electrons on the surface of a torus was also undertaken.