  We present a multi-variable regression analysis to several observable
quantities for a sample of 15 gamma-ray bursts (GRBs). The observables used in
the analysis includes the isotropic gamma-ray energy (E_{gamma,iso}), the peak
energy of the nu F_nu spectrum in the rest frame (E_p'), and the rest frame
break time of the optical afterglow light curves(t_b'). A strong dependence of
E_{gamma, iso on E_p' and t_b' is derived. We also extend the analysis to the
isotropic afterglow energies in the X-ray and the optical bands, respectively,
and find that they are essentially not correlated with E_p' and t_b'. Regarding
the E_{gamma, iso(E_p', t_b') relationship as a luminosity indicator, we
explore the possible constraints on the cosmological parameters using the GRB
sample. Our results indicate 0.13<omega_M<0.49 and 0.50<Omega_Lambda<0.85 for a
flat Universe. The decelerating factor and its cosmological evolution, and the
transition redshift are also investigated. Through Monte Carlo simulations, we
find that the GRB sample satisfying our relationship observationally tends to
be a soft and bright one, and that the contraints on the cosmological
parameters can be much improved either by the enlargement of the sample size or
by the increase of the observational precision. Furthermore, the detections of
a few high redshift GRBs satisfying the correlation could greatly tighten the
constraints (abridged).
