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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2411.18972 |
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| _version_ | 1866911374218625024 |
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| author | Liu, Ricky Ini Tang, Michael |
| author_facet | Liu, Ricky Ini Tang, Michael |
| contents | The generalized degree polynomial $\mathbf{G}_T(x,y,z)$ of a tree $T$ is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that $\mathbf{G}_T$ is determined linearly by the chromatic symmetric function $\mathbf{X}_T$, introduced by Stanley. We present several classes of information about $T$ that can be recovered from $\mathbf{G}_T$ and hence also from $\mathbf{X}_T$. Examples of such information include the double-degree sequence of $T$, which enumerates edges of $T$ by the pair of degrees of their endpoints, and the leaf adjacency sequence of $T$, which enumerates vertices of $T$ by degree and number of adjacent leaves. We also discuss a further generalization of $\mathbf{G}_T$ that enumerates tuples of vertex sets and show that this is also determined by $\mathbf{X}_T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18972 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized degree polynomials of trees Liu, Ricky Ini Tang, Michael Combinatorics The generalized degree polynomial $\mathbf{G}_T(x,y,z)$ of a tree $T$ is an invariant introduced by Crew that enumerates subsets of vertices by size and number of internal and boundary edges. Aliste-Prieto et al. proved that $\mathbf{G}_T$ is determined linearly by the chromatic symmetric function $\mathbf{X}_T$, introduced by Stanley. We present several classes of information about $T$ that can be recovered from $\mathbf{G}_T$ and hence also from $\mathbf{X}_T$. Examples of such information include the double-degree sequence of $T$, which enumerates edges of $T$ by the pair of degrees of their endpoints, and the leaf adjacency sequence of $T$, which enumerates vertices of $T$ by degree and number of adjacent leaves. We also discuss a further generalization of $\mathbf{G}_T$ that enumerates tuples of vertex sets and show that this is also determined by $\mathbf{X}_T$. |
| title | Generalized degree polynomials of trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.18972 |