The number of realisations of a random graph
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914578398445568 |
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| author | Dewar, Sean Nixon, Anthony Smith, Ben |
| author_facet | Dewar, Sean Nixon, Anthony Smith, Ben |
| contents | Determining the number of realisations of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that the $d$-dimensional realisation number of an Erdős-Renyi random graph is either infinity or a power of 2 with exponent computable in polynomial time. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ PSD matrix completion problem with randomly-selected non-diagonal unknown entries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18487 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The number of realisations of a random graph Dewar, Sean Nixon, Anthony Smith, Ben Combinatorics 52C25, 68R12, 14C17 Determining the number of realisations of a graph for a specific choice of edge lengths is a fundamental problem in discrete geometry. In this article we prove that the $d$-dimensional realisation number of an Erdős-Renyi random graph is either infinity or a power of 2 with exponent computable in polynomial time. We also determine a similar formula for the number of complex solutions to the generic rank-$d$ PSD matrix completion problem with randomly-selected non-diagonal unknown entries. |
| title | The number of realisations of a random graph |
| topic | Combinatorics 52C25, 68R12, 14C17 |
| url | https://arxiv.org/abs/2605.18487 |