In this work, we present a novel Position-Sensitive Silicon Photomultiplier based on a charge sharing architecture, which we call Linearly-Graded SiPM (LG-SiPM). We produced both 4x4 mm2 and 8x8 mm2 2D LG-SiPM prototypes with this scheme. The LG-SiPM provides high resolution, low distortion and a very fast output signal, which is fundamental for obtaining a good timing performance. To verify the spatial resolution of the device, we coupled the 4x4 mm2 LG-SiPM prototype to a 4x4 mm2 pixelated LYSO scintillator, with a pitch of 0.8 mm and a height of 10 mm. Under uniform 511 keV irradiation, we were able to easily distinguish the crystal pixels in the flood map and to measure an energy resolution of 18 %.