  Evidence for an extraterrestrial flux of high-energy neutrinos has now been
found in multiple searches with the IceCube detector. The first solid evidence
was provided by a search for neutrino events with deposited energies
$\gtrsim30$ TeV and interaction vertices inside the instrumented volume. Recent
analyses suggest that the extraterrestrial flux extends to lower energies and
is also visible with throughgoing, $\nu_\mu$-induced tracks from the Northern
hemisphere. Here, we combine the results from six different IceCube searches
for astrophysical neutrinos in a maximum-likelihood analysis. The combined
event sample features high-statistics samples of shower-like and track-like
events. The data are fit in up to three observables: energy, zenith angle and
event topology. Assuming the astrophysical neutrino flux to be isotropic and to
consist of equal flavors at Earth, the all-flavor spectrum with neutrino
energies between 25 TeV and 2.8 PeV is well described by an unbroken power law
with best-fit spectral index $-2.50\pm0.09$ and a flux at 100 TeV of
$\left(6.7_{-1.2}^{+1.1}\right)\cdot10^{-18}\,\mathrm{GeV}^{-1}\mathrm{s}^{-1}\mathrm{sr}^{-1}\mathrm{cm}^{-2}$.
Under the same assumptions, an unbroken power law with index $-2$ is disfavored
with a significance of 3.8 $\sigma$ ($p=0.0066\%$) with respect to the best
fit. This significance is reduced to 2.1 $\sigma$ ($p=1.7\%$) if instead we
compare the best fit to a spectrum with index $-2$ that has an exponential
cut-off at high energies. Allowing the electron neutrino flux to deviate from
the other two flavors, we find a $\nu_e$ fraction of $0.18\pm0.11$ at Earth.
The sole production of electron neutrinos, which would be characteristic of
neutron-decay dominated sources, is rejected with a significance of 3.6
$\sigma$ ($p=0.014\%$).
