  We investigate collective spin excitations in graphene with proximity-induced
spin-orbit coupling (SOC) of the Rashba and valley-Zeeman types, as it is the
case, e.g., for graphene on transition- metal-dichalcogenide substrates. It is
shown that, even in the absence of an external magnetic field, such a system
supports collective modes, which correspond to coupled oscillations of the
uniform and valley-staggered magnetizations. These modes can be detected via
both zero-field electron spin resonance (ESR) and zero-field electric-dipole
spin resonance (EDSR), with EDSR response coming solely from Rashba SOC. We
analyze the effect of electron-electron interaction within the Fermi- liquid
kinetic equation and show that the interaction splits both the ESR and EDSR
peaks into two. The magnitude of splitting and the relative weights of the
resonances can be used to extract the spin-orbit coupling constants and
many-body interaction parameters that may not be accessible by other methods.
