Using broken-symmetry hybrid density functional theory, we investigate the relationship between the diradical character and the second hyperpolarizabilities, γ, for a trigonal graphene nanoflake (trigonal GNF) and two types of GNFs with rhombic and bow-tie structures composed of two trigonal GNFs. It is shown that the GNF architecture has a large impact on the diradical character and subsequently on γ. In particular, γ of rhombic GNF with intermediate diradical character is found to be more than twice as large as that of bow-tie GNF with nearly pure diradical character. Then, switching from the singlet to triplet state rhombic GNF exhibits a significant reduction of γ.