State-of-the-art earthquake early warning systems use the early records of seismic waves to estimate the magnitude and location of the seismic source before the shaking and the tsunami strike. Because of the inherent properties of early seismic records, those systems systematically underestimate the magnitude of large events, which results in catastrophic underestimation of the subsequent tsunamis. Prompt elastogravity signals (PEGS) are low-amplitude, light-speed signals emitted by earthquakes, which are highly sensitive to both their magnitude and focal mechanism. Detected before traditional seismic waves, PEGS have the potential to produce unsaturated magnitude estimates faster than state-of-the-art systems. Accurate instantaneous tracking of large earthquake magnitude using PEGS has been proven possible through the use of a Convolutional Neural Network (CNN). However, the CNN architecture is sub-optimal as it does not allow to capture the geometry of the problem. To address this limitation, we design PEGSGraph, a novel deep learning model relying on a Graph Neural Network (GNN) architecture. PEGSGraph accurately estimates the magnitude of synthetic earthquakes down to Mw 7.6-7.7 and determines their focal mechanisms (thrust, strike-slip or normal faulting) within 70 seconds of the event’s onset, offering crucial information for predicting potential tsunami wave amplitudes. Our comparative analysis on Alaska and Western Canada data shows that the GNN outperforms the CNN, especially on test samples with low signal-to-noise ratios, providing more reliable rapid magnitude estimates and enhancing tsunami warning reliability.