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arXiv · 2610.16266

Towards a Sum-of-Squares Proof for the Nonexistence of Conway's 99-graph

Joseph Natal·Oleksiy Al-saadi
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Abstract

Conway's $99$-graph is a strongly regular graph with parameters $(v,k,λ,μ) = (99,14,1,2)$ and whose existence has long been an open problem. We use both a sum-of-squares program and the fact that a strongly regular graph with $λ= 1, μ= 2$ has no diamond or $K_{2,3}$ subgraph to derive a loose upper bound for the number of triangles that such a graph could contain. This bound is less than the number of triangles that the graph's spectrum asserts, implying that it does not exist.

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BibTeXRIS

Joseph Natal, Oleksiy Al-saadi. 2026-10-12. Towards a Sum-of-Squares Proof for the Nonexistence of Conway's 99-graph. https://arxiv.org/abs/2610.16266

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