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| Format: | Preprint |
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2022
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| Online-Zugang: | https://arxiv.org/abs/2211.07409 |
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| _version_ | 1866917640173256704 |
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| author | Narayanan, N. Sawant, Sagar S. |
| author_facet | Narayanan, N. Sawant, Sagar S. |
| contents | The $B$-polynomial defined by J. Awan and O. Bernardi is a generalization of Tutte Polynomial to digraphs. In this paper, we solve an open question raised by J. Awan and O. Bernardi regarding the expansion of $B$-polynomial in elementary symmetric polynomials. We show that the quasisymmetric generalization of the $B$-polynomial distinguishes a class of oriented proper caterpillars and the class of oriented paths. We present a recurrence relation for the quasisymmetric $B$-polynomial involving the deletion of a source or a sink. As a consequence, we prove that a class of digraph $\mathcal{D}$ is distinguishable if and only if the class $\mathcal{D}^{\vee}$ obtained by taking directed join of $K_1$ with each digraph in $\mathcal{D}$ is distinguishable, which concludes that the digraph analogue of Stanley's Tree conjecture holds for a large class of acyclic digraphs. We further study the symmetric properties of the quasisymmetric $B$-polynomial and its relation with certain digraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_07409 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On distinguishing digraphs by its quasisymmetric B-polynomial Narayanan, N. Sawant, Sagar S. Combinatorics 05C15, 05C20, 05C31, 05E05 The $B$-polynomial defined by J. Awan and O. Bernardi is a generalization of Tutte Polynomial to digraphs. In this paper, we solve an open question raised by J. Awan and O. Bernardi regarding the expansion of $B$-polynomial in elementary symmetric polynomials. We show that the quasisymmetric generalization of the $B$-polynomial distinguishes a class of oriented proper caterpillars and the class of oriented paths. We present a recurrence relation for the quasisymmetric $B$-polynomial involving the deletion of a source or a sink. As a consequence, we prove that a class of digraph $\mathcal{D}$ is distinguishable if and only if the class $\mathcal{D}^{\vee}$ obtained by taking directed join of $K_1$ with each digraph in $\mathcal{D}$ is distinguishable, which concludes that the digraph analogue of Stanley's Tree conjecture holds for a large class of acyclic digraphs. We further study the symmetric properties of the quasisymmetric $B$-polynomial and its relation with certain digraphs. |
| title | On distinguishing digraphs by its quasisymmetric B-polynomial |
| topic | Combinatorics 05C15, 05C20, 05C31, 05E05 |
| url | https://arxiv.org/abs/2211.07409 |