Higher matrix-tree theorems and Bernardi polynomial

Abstract

This paper is a continuation of [2]. 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 [1]. A particular case of this family (Corollary 1.6) is the higher-degree generalization of the matrixtree theorem proved in [2], 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).