Existence of strongly regular graphs for the remaining lambda-one, mu-two parameter sets

Determine whether strongly regular graphs exist with parameter sets $(99,14,1,2)$, $(6273,112,1,2)$, and $(494019,994,1,2)$, thereby resolving the corresponding unresolved existence questions, including Conway’s 99-graph problem for $(99,14,1,2)$.

Background

The paper identifies five possible parameter sets for strongly regular graphs with λ=1\lambda=1 and μ=2\mu=2. Two of these, (9,4,1,2)(9,4,1,2) and (243,22,1,2)(243,22,1,2), are known to exist, while the existence of three others is unresolved. The parameter set (99,14,1,2)(99,14,1,2) is especially prominent as Conway’s 99-graph problem. The paper derives the spectra that the associated 3-clique graphs would have if the unresolved strongly regular graphs existed, but it does not settle their existence.

References

A long standing question in algebraic graph theory pertains to the existence of strongly regular graphs with parameters (99,14,1,2), (6273,112,1,2), and (494019,994,1,2). The existence question of a strongly regular graph with parameters (99,14,1,2) is called Conway's 99-graph problem. If such graphs exist then they would all be regular and locally linear, we can then apply Corollary \ref{cor:ll3} giving us the spectrum of their $3$-clique graphs as follows.

— On Clique Graphs and Clique Regular Graphs  (2502.17845 - Petro et al., 25 Feb 2025) in Section 4.5, Example 1,2s

For instance, the possible existence of a graph with parameters $\mathrm{srg}(99,14,1,2)$ has remained open for decades .

— Induced Subgraphs of Order Seven and Their Frequencies in $srg(n,k,1,2)$  (2608.19410 - Reimbayev, 19 Aug 2026) in Section 1, Introduction