  Long duration Gamma-Ray Bursts (LGRBs) may serve as standard candles to
constrain cosmological parameters by probing the Hubble diagram well beyond the
range of redshift currently accessible using type-Ia supernovae. The
standardization of GRBs is based on phenomenological relations between two or
more parameters found from spectral modeling, of which one is strongly
dependent on the cosmological model. The Amati relation links the source-frame
energy ${E_{\mathrm{i,p}}}$ at which the prompt gamma-ray spectral energy
distribution $\nu F_\nu$ peaks, and the isotropic-equivalent bolometric energy
${E_{\mathrm{iso}}}$ emitted during the prompt phase. We performed spectral
analysis of 26 GRBs with known redshift that have been detected by the
Fermi-Large Area Telescope (LAT) during its nine years of operations from July
2008 to September 2017, thus extending the computation of ${E_{\mathrm{iso}}}$
to the 100 MeV range. Multiple components are required to fit the spectra of a
number of GRBs. We found that the Amati relation is satisfied by the 25 LGRBs,
with best fit parameters similar to previous studies that used data from
different satellite experiments, while the only short GRB with known redshift
is an outlier. Using the Amati relation we extend the Hubble diagram to
redshift 4.35 and constrain the Hubble constant and dark-energy density in the
$\Lambda$CDM model, with Fermi-LAT GRBs alone and together with another sample
of 94 GRBs and with the latest Supernovae type-Ia data. Our results are
consistent with the currently acceptable ranges of those cosmological
parameters within errors.
