  In this paper, we propose a model to investigate the electron thermionic
emission from a single-layer graphene (ignoring the effects of substrate) and
to explore its application as the emitter of thermionic energy convertor (TIC).
An analytical formula has been derived, which is a function of temperature,
work function and Fermi energy level. The formula is significantly different
from the traditional Richardson-Dushman (RD) law for which it is independent of
mass to account for the supply function of the electrons in the graphene
behaving like massless Fermion quasiparticles. By comparing with a recent
experiment [Kaili Jiang et al., Nano Research 7, 553 (2014)] measuring electron
thermionic emission from a suspended single layer graphene, our model predicts
that the intrinsic work function of a single-layer graphene is about 4.514 eV
with a Fermi energy level of 0.083 eV. For a given work function, a new scaling
of $T^{3}$ is predicted, which is different from the traditional RD scaling of
$T^2$. If the work function of the graphene is lowered to 2.5 to 3 eV, and the
Fermi energy level is increased to 0.8 to 0.9 eV, it is possible to design a
graphene cathode based TIC operating at around 900 K or lower, as compared with
the metal-based cathode TIC (operating at about 1500 K). With a graphene based
cathode (work function = 4.514 eV) at 900 K, and a metallic based anode (work
function = 2.5 eV) like LaB$_6$ at 425 K, the efficiency of our proposed-TIC is
about 45$\%$
