A cospectral construction for the generalized distance matrix
Fuente:
arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866913600571965440 |
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| author | Friesen, Ori Kolko, Cecily Layman, Nick Lorenzen, Kate Zaske, Sarah Zeigler, Amy |
| author_facet | Friesen, Ori Kolko, Cecily Layman, Nick Lorenzen, Kate Zaske, Sarah Zeigler, Amy |
| contents | The generalized distance matrix of a graph is a matrix in which the $(i,j)$th entry is a function, $f$, of the distance between vertex $i$ and vertex $j$. Depending on the choice of $f$, this family of matrices includes both the adjacency matrix and the traditional distance matrix. We present a cospectral construction for the generalized distance matrix akin to Godsil-McKay Switching. We also investigate a special case of the generalized distance matrix: the exponential distance matrix, which is a matrix where every entry is a value $q$ raised to the power of the distance between the vertices. We give an upper bound on the values of $q$ needed to show a pair of graphs is cospectral for all values of $q$ corresponding to the diameter of the graphs.
We also give cospectral constructions unique to value $q=1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_05389 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A cospectral construction for the generalized distance matrix Friesen, Ori Kolko, Cecily Layman, Nick Lorenzen, Kate Zaske, Sarah Zeigler, Amy Combinatorics The generalized distance matrix of a graph is a matrix in which the $(i,j)$th entry is a function, $f$, of the distance between vertex $i$ and vertex $j$. Depending on the choice of $f$, this family of matrices includes both the adjacency matrix and the traditional distance matrix. We present a cospectral construction for the generalized distance matrix akin to Godsil-McKay Switching. We also investigate a special case of the generalized distance matrix: the exponential distance matrix, which is a matrix where every entry is a value $q$ raised to the power of the distance between the vertices. We give an upper bound on the values of $q$ needed to show a pair of graphs is cospectral for all values of $q$ corresponding to the diameter of the graphs. We also give cospectral constructions unique to value $q=1/2$. |
| title | A cospectral construction for the generalized distance matrix |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.05389 |