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arXiv · 2504.02449

Strongly regular graphs with parameters (85,14,3,2) do not exist

Sergey Shpectorov·Tianxiao Zhao
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Abstract

We investigate the second smallest unresolved feasible set of parameters of strongly regular graphs, $(v,k,\lambda,\mu)=(85,14,3,2)$. Using the classification of cubic graphs of small degree, we restrict possible local structure of such a graph $G$. After that, we exhaustively enumerate possible neighbourhoods of a maximal $3$-clique of $G$ and check them against a variety of conditions, including the combinatorial ones, coming from $\lambda=3$ and $\mu=2$, as well as the linear algebra ones, utilising the Euclidean representation of $G$. These conditions yield contradiction in all cases, and hence, no $\mathrm{srg}(85,14,3,2)$ exists.

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Sergey Shpectorov, Tianxiao Zhao. 2025-04-03. Strongly regular graphs with parameters (85,14,3,2) do not exist. https://arxiv.org/abs/2504.02449

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