An audit of published extremal results for topological graph indices: counterexamples, certificates, and repairs

The literature on topological graph indices contains many extremal results: statements of the form “over the graph class Gn,π the index I is maximized (minimized) exactly by the graphs Hn,π, with value φ(n,π)”. We describe a reproducible audit methodology: published extremal claims are transcribed into a formal, machine-readable claim model and tested by exhaustive enumeration over all graphs of small order, with exact arithmetic certificates for every reported contradiction. Auditing 1271 machine-encoded claim rows (1238 transcribed from over 140 published papers, plus 33 calibration and engine cross-check rows), we document 24 defect cases in 23 papers (2015–2027), none of which had a paper-specific published correction as of 11 August 2026. One of them reinstates a statement whose corrected form had been in print for a decade. Each case comes with explicit counterexamples and exact certificates; where possible we prove complete repaired theorems, including sharp corrected bounds with full equality characterizations. Recurring failure mechanisms are identified — unproved “analogous” branches, boundary parameter values excluded by the proof but included in the statement, incomplete equality characterizations, empty extremal families, and formula–proof inconsistencies — together with three documented propagation chains (reuse, silent correction, reintroduction). Two further defects found by the same pipeline turned out to have been corrected in the literature; they are reported separately as retrospective validation of the method. Contents: the preprint (30 pages) and the frozen data-and-code archive (topological_index_audit_archive.zip): 1271 machine-encoded claim rows in 194 JSON claim files, the evaluation engine with self-test (validated against OEIS A001349, A000055, A001429), a mutation test (90 of 90 sampled mutants flipped), an independent per-case verifier, an exhaustive connected-bipartite generator for even orders up to 8, the full verification output at the freeze date (1214 PASS, 51 FAIL, 6 EMPTY, 0 unexpected, every FAIL and EMPTY accounted for), a transcription-error log, a per-case map from the paper's master-table rows to claim files, a README with reproduction commands, and SHA-256 checksums. Requirements: Python 3.11+ with networkx ≥ 3.3. Audit freeze date: 11 August 2026. Publisher PDFs of the audited papers are not included (copyright).