Graffiti 154 under spectral standard-deviation conventions
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Is the standard deviation of a graph’s adjacency eigenvalues at most its vertex count divided by average distance? This is the standard-deviation reading of Graffiti Conjecture 154; alternative meanings of “deviation” change the target.
Why it matters and what counts as progress
Why it matters: A counterexample would refute the standard-deviation interpretation while leaving differently defined notions of spectral deviation separate.
What counts as progress: Certify a counterexample under an explicit average-distance convention or prove the inequality for a restricted graph family.
How the problem developed
- July 2004 · Conjecture [1]
Siemion Fajtlowicz Listed the bound that deviation of eigenvalues is at most graph order divided by average distance.
- 11 Jun 2026 · Counterexample [1]
Repository result: exact counterexample certificates Reported connected lollipop counterexamples to the standard-deviation reading under both average-distance conventions, with the first listed violation at order 118.
AI contribution
AI-found graphs refute Graffiti 154 under a standard-deviation reading
Lollipop graphs with 118 or 120 vertices violate Graffiti Conjecture 154 when “deviation” means the standard deviation of adjacency eigenvalues. A 144-vertex graph violates it under both standard average-distance conventions.
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AI-found graphs refute Graffiti 154 under a standard-deviation reading