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Magnetic nanoribbons based on one-dimensional materials are potential candidates for spin caloritronics devices. Here, we constructed ferromagnetic graphene nanoribbons with zigzag and Klein edges (N-ZKGNRs, N = 4–21) and found that the N-ZKGNRs are in the indirect-gap bipolar magnetic semiconducting state (BMS). Moreover, when a temperature difference is applied through the nanoribbons, spin-dependent currents with opposite flow directions and opposite spin directions are generated, indicating the occurrence of the spin-dependent Seebeck effect (SDSE). In addition, the spin-dependent Seebeck diode effect (SDSD) also appeared in these devices. More importantly, we found that the BMS with a larger bandgap is promising for generating the SDSD, while the BMS with a smaller bandgap is promising for generating the SDSE. These findings show that ZKGNRs are promising candidates for spin caloritronics devices.