The well-known Amati and Yonetoku relations in gamma-ray bursts show strong correlations between the rest-frame νfν spectrum peak energy, Ep,i, and the isotropic energy, Eiso, as well as isotropic peak luminosity, Liso. Recently, Peng et al. showed that the cosmological rest-frame spectral widths are also correlated with Eiso and with Liso. In this paper, we select a sample including 141 BEST time-integrated F spectra and 145 BEST peak flux P spectra observed by Konus–Wind with known redshift to recheck the connection between the spectral width and Eiso as well as Liso. We define six types of absolute spectral widths as the differences between the upper (E2) and lower energy bounds (E1) of the full width at 50%, 75%, 85%, 90%, 95%, and 99% of maximum of the EFE versus E spectra. It is found that all of the rest-frame absolute spectral widths are strongly positively correlated with Eiso as well as Liso for the long burst for both the F and P spectra. All of the short bursts are outliers for the width–Eiso relation, and most of the short bursts are consistent with the long bursts for the width–Liso relation for both F and P spectra. Moreover, all of the location energies, E2 and E1, corresponding to various spectral widths, are also positively correlated with Eiso as well as Liso. We compare all of the relations with the Amati and Yonetoku relations and find that the width–Eiso and width–Liso relations, when the widths are at about 90% maximum of the EFE spectra, almost overlap with the Amati relation and the Yonetoku relation, respectively. The correlations of E2 − Eiso, E1 − Eiso and E2 − Liso, E1 − Liso when the location energies are at 99% of maximum of the EFE spectra are very close to the Amati and Yonetoku relations, respectively. Therefore, we confirm the existence of tight width–Eiso and width–Liso relations for long bursts. We further show that the spectral shape is indeed related to Eiso and Liso. The Amati and Yonetoku relations are not necessarily the best relationships for relating the energy to the Eiso and Liso. They may be special cases of the width–Eiso and width–Liso relations or the energy–Eiso and energy–Liso relations.