Two-photon absorption (TPA) properties of two trefoil-shaped compounds with different edge linkages--tris(hexadehydrotribenzo[12]annulene) and tris(tetradehydrotribenzo[12]annulene)--and three asterisk-shaped compounds having no edge-linkage--hexakis(phenylethynyl)benzenes--are investigated experimentally by the open-aperture Z-scan and TPA-induced fluorescence methods with wavelength tuneable femtosecond pulses. The compound with ethynylene edge-linkage exhibits the most intense TPA (the maximal TPA cross section is 1300+/-170 GM at 572 nm where 1 GM=10(-50) cm(4) s molecule(-1) photon(-1)). The TPA activity of the compounds is primarily explained in terms of the planarity of the molecules in relation with the type of edge-linkage.