The manuscript presents the design and investigation of square and L-shaped graphene plasmonic nano-antenna arrays on a silicon dioxide substrate for plasmonic biosensing applications. The design parameters, such as the shape of the nanostructure, spacing between the antenna array elements, size of the antenna array and chemical potential of the graphene are used to analyze the plasmonic resonance characteristics of the proposed plasmonic nano-antenna array. Initially, dispersive characteristics of graphene, such as conductivity, permittivity and refractive index are analyzed using its chemical potential for the validation of the graphene material for plasmonic sensing applications. The proposed square and L-shaped nanopatch antenna structures are radiating at 30 THz with a reflection coefficient of -28.4 dB and − 44.6 dB respectively at 0.1 eV chemical potential. It is observed that graphene nanopatch antenna radiates at 30 THz, 115 THz and 176 THz at a chemical potential of 1.3 eV. Further, the study demonstrated that the designed square and L-shaped nano-antenna exhibit a gain of 3.52 dB and 9.51 dB respectively at 30 THz. The gain can be increased further using a square and L-shaped nano-antenna array of dimensions 2 × 2, 3 × 3 and 4 × 4. It is observed that maximum gains of 33.44 dB and 30.86 dB are obtained for the square and L-shaped 4 × 4 plasmonic nanoarray antenna respectively. Finally, a nanocircuit model of square and L-shaped nano-antennas is proposed and validated through CST simulations.