  Massive gravitons are features of some alternatives to general relativity.
This has motivated experiments and observations that, so far, have been
consistent with the zero mass graviton of general relativity, but further tests
will be valuable. A basis for new tests may be the high sensitivity
gravitational wave experiments that are now being performed, and the higher
sensitivity experiments that are being planned. In these experiments it should
be feasible to detect low levels of dispersion due to nonzero graviton mass.
One of the most promising techniques for such a detection may be the pulsar
timing program that is sensitive to nano-Hertz gravitational waves.
  Here we present some details of such a detection scheme. The pulsar timing
response to a gravitational wave background with the massive graviton is
calculated, and the algorithm to detect the massive graviton is presented. We
conclude that, with 90% probability, massles gravitons can be distinguished
from gravitons heavier than $3\times 10^{-22}$ eV (Compton wave length
$\lambda_{\rm g}=4.1 \times 10^{12}$ km), if biweekly observation of 60 pulsars
are performed for 5 years with pulsar RMS timing accuracy of 100 ns. If 60
pulsars are observed for 10 years with the same accuracy, the detectable
graviton mass is reduced to $5\times 10^{-23}$ eV ($\lambda_{\rm g}=2.5 \times
10^{13}$ km); for 5-year observations of 100 or 300 pulsars, the sensitivity is
respectively $2.5\times 10^{-22}$ ($\lambda_{\rm g}=5.0\times 10^{12}$ km) and
$10^{-22}$ eV ($\lambda_{\rm g}=1.2\times 10^{13}$ km). Finally, a 10-year
observation of 300 pulsars with 100 ns timing accuracy would probe graviton
masses down to $3\times 10^{-23}$ eV ($\lambda_{\rm g}=4.1\times 10^{13}$ km).
