Exact Enumeration of Connected 4‑Regular Multigraphs up to 6 Vertices: Vacuum Feynman Diagrams in Scalar φ⁴ Theory

We present the complete enumeration of connected 4‑regular multigraphs on n vertices for n = 1,…,6, which correspond to connected vacuum Feynman diagrams in scalar φ⁴ quantum field theory. For tadpole‑free diagrams (no self‑loops) the counts are: 1, 1, 3, 6, 19 for n = 2,…,6. For all vacuum diagrams (self‑loops allowed) the counts are: 1, 2, 4, 10, 28, 97 for n = 1,…,6. For each graph we compute the order of its automorphism group (the symmetry factor). The results resolve long‑standing discrepancies in the literature (e.g., 70/71/73 vs our 19 for n=6 without self‑loops). The enumeration is performed by exhaustive backtracking with isomorphism rejection, verified independently via Wick contraction counting up to n=4 and by algebraic analysis for n=4. The Python source code is provided as supplementary material.