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

Conway’s $99$-graph is a strongly regular graph with parameters $(v,k,\lambda ,\mu )=(99,14,1,2)$ 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 $\lambda =1,\mu =2$ 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.