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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.02449 |
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| _version_ | 1866916672791642112 |
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| author | Shpectorov, Sergey Zhao, Tianxiao |
| author_facet | Shpectorov, Sergey Zhao, Tianxiao |
| contents | We investigate the second smallest unresolved feasible set of parameters of strongly regular graphs, $(v,k,λ,μ)=(85,14,3,2)$. 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 $λ=3$ and $μ=2$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02449 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly regular graphs with parameters (85,14,3,2) do not exist Shpectorov, Sergey Zhao, Tianxiao Combinatorics 05E30 We investigate the second smallest unresolved feasible set of parameters of strongly regular graphs, $(v,k,λ,μ)=(85,14,3,2)$. 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 $λ=3$ and $μ=2$, 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. |
| title | Strongly regular graphs with parameters (85,14,3,2) do not exist |
| topic | Combinatorics 05E30 |
| url | https://arxiv.org/abs/2504.02449 |