On distinguishing digraphs by its quasisymmetric B-polynomial

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.