  Binary neutron star mergers are strong gravitational wave (GW) sources and
the leading candidates to interpret short duration gamma-ray bursts (SGRBs).
Under the assumptions that SGRBs are produced by double neutron star mergers
and that the X-ray plateau followed by a steep decay as observed in SGRB X-ray
light curves marks the collapse of a supra-massive neutron star to a black hole
(BH), we use the statistical observational properties of {\em Swift} SGRBs and
the mass distribution of Galactic double neutron star systems to place
constraints on the neutron star equation of state (EoS) and the properties of
the post-merger product. We show that current observations already put
following interesting constraints: 1) A neutron star EoS with a maximum mass
close to a parameterization of $M_{\rm max} = 2.37\,M_\odot
(1+1.58\times10^{-10} P^{-2.84})$ is favored; 2) The fractions for the several
outcomes of NS-NS mergers are as follows: $\sim40\%$ prompt BHs, $\sim30\%$
supra-massive NSs that collapse to BHs in a range of delay time scales, and
$\sim30\%$ stable NSs that never collapse; 3) The initial spin of the newly
born supra-massive NSs should be near the breakup limit ($P_i\sim1 {\rm ms}$),
which is consistent with the merger scenario; 4) The surface magnetic field of
the merger products is typically $\sim 10^{15}$ G; 5) The ellipticity of the
supra-massive NSs is $\epsilon \sim (0.004 - 0.007)$, so that strong GW
radiation is released post the merger; 6) Even though the initial spin energy
of the merger product is similar, the final energy output of the merger product
that goes into the electromagnetic channel varies in a wide range from several
$10^{49}$ erg to several $10^{52}$ erg, since a good fraction of spin energy is
either released in the form of GW or falls into the black hole as the
supra-massive NS collapses.
