Complete one-loop connected cubic three-point graphs

For a real phi cubed theory with labelled external points, the connected three-point function at third order in the coupling constant has seven labelled one-loop Feynman diagrams: one triangle, three external-branch bubble insertions, and three tadpole-ended chains. The symmetry factors are respectively $1,2,2$, with graph weight $1/S$. To prove completeness, use $3V=2I+3$ and $L=I-V+1$: $V=3$ gives $I=3$ and $L=1$. The unique internal cycle is then a triangle, a pair of parallel edges, or a self-loop; the degree-three condition fixes the external attachments. Only the triangle remains one-particle irreducible after amputation of external propagators. A tadpole-free prescription removes the three chains but must be stated separately.