Density functional theory (DFT) calculations were employed to investigate the electronic properties of an H-atom terminated graphene patch (hydrographene) smaller than a rhombic C96H26 structure with zigzag edges. Depending on shapes and sizes of hydrographenes, some hydrographenes have the triplet ground state where unpaired electrons are localized on their zigzag edges. The stability of the triplet spin state is diminished, decreasing the hydrographene sizes. The existence of the localized spin densities allows triplet dioxgen to bind into a hydrographene. According to the DFT calculations, the energetics of the dioxygen bindings is negatively influenced by downsizing hydrographenes, as well as depends on their shapes. The size-and shape-dependences of the dioxygen bindings reflect from the stability of the triplet state of a hydrographene, because its localized unpaired electrons can be utilized to be attached to an unpaired electron of triplet dioxygen. Density functional theory (DFT) calculations were employed to investigate the electronic properties of an H-atom terminated graphene patch (hydrographene) smaller than a rhombic C96H26 structure with zigzag edges. According to shapes and sizes of hydrographenes, some hydrographenes have the triplet ground state where unpaired electrons are localized on their zigzag edges. The stability of the triplet spin state is diminished, decreasing the hydrographene sizes. The existence of the localized spin densities allows triplet dioxgen to bind into a hydrographene. According to the DFT calculations, the energetics of the dioxygen bindings is negatively influenced by downsizing hydrographenes, as well as depends on their shapes. The size-and shape-dependences of the dioxygen bindings reflect from the stability of the triplet state of a hydrographene, because its localized unpaired electrons can be utilized to be attached to an unpaired electron of triplet dioxygen.