There exist graphs of which all detours are Hamiltonian paths. Such graphs are called by Diudea (J. Chem. Inf. Comput. Sci. 1997, 37, 1101-1108) full Hamiltonian detour FHΔ graphs. These graphs show the maximal value of the Detour index while the minimal value of the Cluj-Detour index. A selected set of cubic graphs on 16 vertices is tested for the distribution of the relative centrality RC val...

It is well known that 3–regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3–regular graphs without reducing the girth, thereby proving that such graphs with arbitrarily large girth also exist. The resulting graphs can be 1–, 2– or 3–edge-connected depending on the construction chosen. From the constructions arise (nai...