This project mathematically models the self-assembly of DNA nanostructures in the shape of select Archimedean graphs using the flexible tile model. Under three different sets of restrictions called scenarios, we employ principles of linear algebra and graph theory to determine the minimum number of different DNA branched molecules and bond types needed to construct the desired shapes, theoretically reducing laboratory costs and the waste of biomaterials. We determine exact values for $T_3(G)$, the minimum number of molecule (or ``tile") types needed for all six order 12 and 24 Archimedean graphs. We also determine exact values for $B_3(G)$, the minimum number of strand (or ``bond-edge") types, for three of the six graphs and establish bounds for the remaining three. Two algorithms, implemented as Python scripts, are used to analyze proposed design strategies for the graphs.