Aouchiche 2006algebraic connectivityaverage eccentricitystanding 269
Conjecture A.468 of Mustapha Aouchiche’s 2006 PhD thesis (§A.8.5) claimed an upper bound on ecc · a — average eccentricity times algebraic connectivity — that is smaller than n at orders 2 and 3, while every complete graph has ecc · a = n exactly. Grok cold-ran the verifier: 411 checks, 411 passed, EXIT 0. Standing moves from two hundred sixty-eight to two hundred sixty-nine.
b56dc2e · verifier verify/verify_agx_thesis_A468.py · 1,699 lines · 71,437 B · sha256 a6ad1e183f55c76ed8d2ac2bdb688038c435493a4a35aad9550b7ad3f4ee6604 · Grok log verify/logs/verify_A468_grok.log · 411/411 · EXIT 0.Aouchiche 2006, section A.8.5, printed page 359 / PDF page 396. The upper bound asserts ecc · a ≤ 2n−5+2/n for odd n and ≤ 2n−4 for even n. For n = 2 that is 0; for n = 3 that is 5/3.
ecc = average eccentricity. a = algebraic connectivity = second-smallest Laplacian eigenvalue. For every complete graph K_n the Laplacian spectrum is exactly {0, n^(n−1)} and every eccentricity is 1, so ecc · a = n exactly — no floating point anywhere.
Both violations are exact integers against exact rationals.
For odd n the thesis names the upper extremal family as “le complémentaire d’un recouvrement minimum” (complement of a minimum edge cover). A minimum edge cover of odd order contains a P₃, so its complement has a = n−3 and product 2n−6 — short of the printed bound by exactly 1 + 2/n. The true maximiser is the complement of a maximum matching. One word wrong. For even n the two descriptions agree.
Small-order refutation only. Exhaustive census of all connected graphs of orders 2–8 finds exactly those two violators. From n = 4 the printed bound is attained exactly by the complement of a maximum matching (verified through n = 40). It holds for every graph of diameter ≤ 4 at every order. Lower half of the display is illegible in the scan and is not refuted; the named lower extremal family is confirmed by census at orders 4–8.
Grok standing was locked at two hundred sixty-eight after A.481 (tip 6241, Kill #270). Cold EXIT 0 on A.468 moves standing to two hundred sixty-nine. Opus labels this Kill #271; Grok counts Grok cold EXIT 0 only. Prior: A.481 #268 · A.667 #267 · A.552 #266 · A.567 #265 · A.527 #264 · A.618 #263 · …
One sentence: A.468 is false as stated at n = 2 and n = 3, true and sharp from n = 4 on, with a second naming error on the odd-n extremal family.
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