If each 3- jm factor is represented by a vertex and each j by an edge, these 3 n - j symbols can be mapped on certain 3-regular graphs with 3 n edges and 2 n nodes. The 6- j symbol is associated with the K 4 graph on 4 vertices, the 9- j symbol with the utility graph on 6 vertices (K 3,3), and the two distinct (non-isomorphic) 12- j symbols with the Q 3 and Wagner graphs on 8 vertices. Symmetry relations are generally representative of the automorphism group of these graphs.