Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909083178631168 |
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| author | Dewar, Sean |
| author_facet | Dewar, Sean |
| contents | A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius characterised exactly which vertex-transitive graphs are globally rigid solely by their degree and maximal clique number, two easily computable parameters for vertex-transitive graphs. In this short note we will extend this characterisation to all graphs that are determined by their automorphism group. We do this by characterising exactly which edge-transitive graphs and distance-regular graphs are globally rigid by their minimal and maximal degrees. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_03965 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane Dewar, Sean Combinatorics 52C25 A graph is said to be globally rigid if almost all embeddings of the graph's vertices in the Euclidean plane will define a system of edge-length equations with a unique (up to isometry) solution. In 2007, Jackson, Servatius and Servatius characterised exactly which vertex-transitive graphs are globally rigid solely by their degree and maximal clique number, two easily computable parameters for vertex-transitive graphs. In this short note we will extend this characterisation to all graphs that are determined by their automorphism group. We do this by characterising exactly which edge-transitive graphs and distance-regular graphs are globally rigid by their minimal and maximal degrees. |
| title | Classifying the globally rigid edge-transitive graphs and distance-regular graphs in the plane |
| topic | Combinatorics 52C25 |
| url | https://arxiv.org/abs/2202.03965 |