On the average squared radius of gyration of a family of embeddings of subdivision graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912048392175616 |
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| author | Cantarella, Jason Schumacher, Henrik Shonkwiler, Clayton |
| author_facet | Cantarella, Jason Schumacher, Henrik Shonkwiler, Clayton |
| contents | Suppose we have an embedding of a graph $\mathbf{G}$ created by subdividing the edges of a simpler graph $\mathbf{G'}$. The edges of $\mathbf{G}$ can be divided into subsets which join pairs of ``junction'' vertices in $\mathbf{G'}$. The displacement vectors of the edges in each subset sum to the displacement between junctions. We can construct a family of embeddings of $\mathbf{G}$ with the same junction positions by rearranging the displacements in each group. In this paper, we show that the average (squared) radius of gyration of these embeddings is given by a simple formula involving a weighted (squared) radius of gyration of the positions of the junctions and the sum of the squares of the lengths of the edges of $\mathbf{G}$ and $\mathbf{G'}$. This ensemble of graph embeddings arises naturally in polymer science. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the average squared radius of gyration of a family of embeddings of subdivision graphs Cantarella, Jason Schumacher, Henrik Shonkwiler, Clayton Combinatorics 05C62, 92E10, 82D60 Suppose we have an embedding of a graph $\mathbf{G}$ created by subdividing the edges of a simpler graph $\mathbf{G'}$. The edges of $\mathbf{G}$ can be divided into subsets which join pairs of ``junction'' vertices in $\mathbf{G'}$. The displacement vectors of the edges in each subset sum to the displacement between junctions. We can construct a family of embeddings of $\mathbf{G}$ with the same junction positions by rearranging the displacements in each group. In this paper, we show that the average (squared) radius of gyration of these embeddings is given by a simple formula involving a weighted (squared) radius of gyration of the positions of the junctions and the sum of the squares of the lengths of the edges of $\mathbf{G}$ and $\mathbf{G'}$. This ensemble of graph embeddings arises naturally in polymer science. |
| title | On the average squared radius of gyration of a family of embeddings of subdivision graphs |
| topic | Combinatorics 05C62, 92E10, 82D60 |
| url | https://arxiv.org/abs/2409.18767 |