As the worldwide incidence of stroke increases, supernumerary robotic limbs (SRLs), more specifically supernumerary robotic fingers (SRFs), present a potentially effective solution for enhancing the task related functionality of the upper-limbs of stroke survivors. This study investigated the impact of an SRF use on cognitive function, employing both electroencephalography (EEG) and graph theory analysis (GTA). Three tasks were undertaken by 30 subjects with and without the SRF. The results revealed notable changes during driving, as well as SRF-assisted activities. Increased clustering coefficients in emotional expression regions during SRF-assisted driving contrasted with reduced connectivity strength between frontal brain regions, suggesting altered attention control. Local efficiency (LE) analysis indicated disruptions in neuronal communication during SRF-assisted tasks, impacting attention, motor planning, and judgment actions. Likewise, statistical analyses demonstrated noteworthy shifts in mental states between phases, emphasizing improved communication efficiency with negative LE values. These findings increased our understanding of SRF influence on brain dynamics, which may help guide future development of assistive tools and technology to assist stroke survivors with their day-to-day functional activities.