关于通过拟对称B多项式区分有向图
On distinguishing digraphs by its quasisymmetric B-polynomial
AI总结：
该研究解决了B多项式初等对称展开的公开问题，证明拟对称B多项式可区分两类有向图，给出其递推关系，推导有向图可区分性的等价条件，验证Stanley树猜想有向版本对大类无环有向图成立，并探究其对称性质。
摘要：
由J. Awan和O. Bernardi定义的B多项式是Tutte多项式在有向图上的推广。本文解决了J. Awan与O. Bernardi提出的关于B多项式在初等对称多项式中展开的公开问题。我们证明了B多项式的拟对称推广可以区分一类有向真毛虫图（oriented proper caterpillars）和有向路径类（oriented paths）。我们给出了拟对称B多项式的一个递推关系，该关系涉及删除源点或汇点的操作。作为推论，我们证明了一类有向图$\u27e8\ud835\udc9f\u27e9$是可区分的，当且仅当对$\u27e8\ud835\udc9f\u27e9$中每个有向图与$K_1$取有向并得到的类$\u27e8\ud835\udc9f^{\u2228}\u27e9$是可区分的，由此得出Stanley树猜想的有向图版本对一大类无环有向图成立。我们进一步研究了拟对称B多项式的对称性质及其与某些有向图的关系。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
