  In this paper we present a correlation between the peak spectral energy of
gamma-ray bursts (GRBs) and the peak bolometric luminosity of the underlying
supernovae (SNe), based on a sample of four pairs of GRBs-SNe with
spectroscopically confirmed connection. Combining it with the well-known
relation between the peak spectral energy and the isotropic equivalent energy
of GRBs, we obtain an upper limit on the isotropic energy of GRBs, which is
\approx 10^{52} erg (L_{SN,peak}/10^{43} erg s^{-1})^{10}, where L_{SN,peak} is
the peak bolometric luminosity of the SNe. Our results suggest that the
critical parameter determining the GRB-SN connection is the peak luminosity of
SNe, rather than the feature of the SN spectra and/or the SN explosion energy
as commonly hypothesized. Since it is generally believed that the peak
luminosity of SNe powered by radioactive decays is related to the amount of
56Ni produced in the SN explosion, the mass of 56Ni may be a key physical
factor for understanding the nature of GRBs and their connection with SNe.
Application of our relation to Type Ibc SNe with normal peak luminosities
indicates that if those normal SNe have GRBs accompanying them, the GRBs would
be extremely soft and sub-energetic in gamma-rays, and hence easier to detect
with X-ray or UV detectors than with gamma-ray detectors.
