Enhancing light-matter interaction is crucial for boosting the performance of nanophotonic devices, which can be achieved via plasmon-induced transparency (PIT). This study introduces what we believe to be a novel E-type metamaterial structure crafted from a single graphene layer. The structure, comprising a longitudinal graphene ribbon and three horizontal graphene strips, leverages destructive interference at terahertz frequencies to manifest triple plasmon-induced transparency (triple-PIT). Through a comparison of simulations using the finite difference time domain (FDTD) method and theoretical coupled-mode calculations, we elucidate the physical mechanism behind triple-PIT. Our analysis shows that the PIT effect arises from the interplay between two single-PITs phenomena, further explored through field distribution studies. Additionally, we investigate the impact of varying Fermi levels and carrier mobility on the transmission spectrum, achieving amplitude modulation in photoelectric switches of 85.5%, 99.2%, and 93.8% at a carrier mobility of 2 m2/(V·s). Moreover, we explore the relationship between Fermi levels and carrier mobility concerning the slow light effect, discovering a potential group index of up to 1021 for the structure. These insights underscore the significant potential of this graphene-based metamaterial structure in enhancing optical switches, modulators, and slow light devices.