  The Tutte polynomial is a fundamental invariant of graphs. In this article,
we define and study a generalization of the Tutte polynomial for directed
graphs, that we name B-polynomial. The B-polynomial has three variables, but
when specialized to the case of graphs (that is, digraphs where arcs come in
pairs with opposite directions), one of the variables becomes redundant and the
B-polynomial is equivalent to the Tutte polynomial. We explore various
properties, expansions, specializations, and generalizations of the
B-polynomial, and try to answer the following questions: 1. what properties of
the digraph can be detected from its B-polynomial (acyclicity, length of paths,
number of strongly connected components, etc.)? 2. which of the marvelous
properties of the Tutte polynomial carry over to the directed graph setting?
The B-polynomial generalizes the strict chromatic polynomial of mixed graphs
introduced by Beck, Bogart and Pham. We also consider a quasisymmetric function
version of the B-polynomial which simultaneously generalizes the Tutte
symmetric function of Stanley and the quasisymmetric chromatic function of
Shareshian and Wachs.
