We investigate the relation between the non-Gaussianities of the
primordial perturbations and the sound speed of the tensor
perturbations, that is, the propagation speed of the gravitational
waves. We find that the sound speed of the tensor perturbations is
directly related not to the auto-bispectrum of the tensor perturbations
but to the cross-bispectrum of the primordial perturbations, especially,
the scalar-tensor-tensor bispectrum. This result is in sharp contrast
with the case of the scalar (curvature) perturbations, where their
reduced sound speed enhances their auto-bispectrum. Our findings
indicate that the scalar-tensor-tensor bispectrum can be a powerful tool
to probe the sound speed of the tensor perturbations.