  We evaluate the nonlinear coefficient of graphene-on-silicon waveguides
through the coincidence measurement of photon-pairs generated via spontaneous
four-wave mixing. We observed the temporal correlation of the photon-pairs from
the waveguides over various transfer layouts of graphene sheets. A simple
analysis of the experimental results using coupled-wave equations revealed that
the atomically-thin graphene sheets enhanced the nonlinearity of silicon
waveguides up to ten-fold. The results indicate that the purely
$\chi^{(3)}$-based effective nonlinear refractive index of graphene is on the
order of $10^{-13}$ m$^2$/W, and provide important insights for applications of
graphene-based nonlinear optics in on-chip nanophotonics.
