From the viewpoint of "diradical character," referred to as "diradicalology," we investigate the second hyperpolarizability γ - the molecular third-order nonlinear optical (NLO) property - of one-dimensional supermolecular systems composed of acetylene-linked phenalenyl/pyrene rings using long-range corrected spin-unrestricted density functional theory. It turns out that the pyrene-based superpolyenes (Py-n) behave like closed-shell systems, whereas phenalenyl-based superpolyenes (Ph1-n and Ph2-n) have different diradical characters depending on the linked form, that is, Ph1-n and Ph2-n have intermediate and pure diradical characters, respectively. In comparison with Py-n and Ph2-n, the longitudinal γ of Ph1-n is significantly larger, and it displays larger enhancement as a function of system size. Substitutions to the terminal rings by donor (NH2) and acceptor (NO2) groups further enhance γ in Ph1-n, more than in Ph2-n and Py-n. These results are in agreement with the structure-property relationships derived for open-shell NLO systems with symmetric and asymmetric charge distributions (Nakano et al., J. Chem. Phys. 2010, 133, 154302).