# Abstract matrix-tree theorem and Bernardi polynomial

## TL;DR

This paper extends the matrix-tree theorem to directed and undirected graphs using a generalized Tutte polynomial, providing new identities and a shorter proof for the higher-degree case.

## Contribution

It introduces a three-parameter family of identities involving a Tutte polynomial variant, generalizing the matrix-tree theorem with new proofs and parallel results for undirected graphs.

## Key findings

- Proves a three-parameter family of identities for a Tutte polynomial variant.
- Provides a shorter proof for the higher-degree matrix-tree theorem.
- Establishes parallel results for undirected graphs.

## Abstract

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).