  This paper is a continuation of arXiv:1612.03873. We prove a three-parameter
family of identities (Theorem 1.1) involving a version of the Tutte polynomial
for directed graphs introduced by Awan and Bernardi in arXiv:1610.01839. A
particular case of this family (Corollary 1.6) is the higher-degree
generalization of the matrix-tree theorem proved in arXiv:1612.03873, which
thus receives a new proof, shorter (and less direct) than the original one. The
theory has a parallel version for undirected graphs (Theorem 1.2).
