Exact dumbbell and lollipop witnesses expose two reading-sensitive failures beyond earlier search horizons.

Graffiti Conjecture 143 is false as stated for connected graphs under both standard conventions for average distance, and Conjecture 154 is false as stated under the standard-deviation reading of “deviation,” again under both conventions. Exact interval and Sturm calculations certify the counterexamples, which lie beyond the search ranges reported in a 2025 survey.

The statements appear in S. Fajtlowicz's July 2004 Written on the Wall compilation and compare eigenvalue statistics with graph size and average distance. Both were listed open in the Roucairol–Cazenave ECAI 2025 survey (arXiv:2409.18626).

The authors were notified privately.

The frozen statements

The glyph-decoded July 2004 compilation states:

• Conjecture 143: “variance of positive eigenvalues ≤ size / average distance.”
• Conjecture 154: “deviation of eigenvalues ≤ n / average distance.”

Conjecture 143

For average distance 2W/n², dumbbell(7,12,20) with n = 39 is a certified counterexample. It also violates the distinct-pairs convention 2W/(n(n−1)). For that second convention, dumbbell(6,12,19) already gives a counterexample at n = 37.

All margins are certified by exact rational interval arithmetic of width ≤ 2⁻⁷⁰. No floating-point computation lies on the acceptance path.

Conjecture 154

Under the standard-deviation reading used by Favaron–Mahéo–Saclé in 1993 and by the only known machine formalization, the printed inequality fails under both average-distance conventions. The violation reduces to the integer inequality 8mW² > n⁵(n−1)², or respectively 8mW² > n⁷.

Certified counterexamples are lollipop(48,70) at n = 118, lollipop(50,70) at n = 120, and lollipop(72,72) at n = 144; the last violates both conventions.

Prior searches

The 2025 survey reports that Conjecture 143 was searched through size 100 by eight algorithms and Conjecture 154 through size 50. Conjecture 154 also survived the 1990–91 BDF attack. The certified violations lie beyond those prior search horizons.

Artifacts

Limitations

No general minimality for Conjecture 143 is claimed below n = 37, although both mainstream average-distance readings are refuted as stated. Conjecture 154 is refuted only under the standard-deviation reading: under the mean-absolute-deviation reading, these instances provably do not violate the inequality. Its minimality claim is confined to the lollipop family.

The results have not been externally refereed. Corrections will be logged, dated, and never silent.