Tutte polynomial and G-parking functions

TL;DR

This paper characterizes external activity in terms of G-parking functions and expresses the Tutte polynomial of a graph using these functions, linking combinatorial properties to polynomial invariants.

Contribution

It introduces the concept of bridge vertices in G-parking functions and provides a new expression for the Tutte polynomial based on these functions.

Findings

Tutte polynomial enumerates G-parking functions by bridge vertices.

Defined bridge vertex for G-parking functions.

Expressed Tutte polynomial in terms of G-parking functions.

Abstract

Let $G$ be a connected graph with vertex set $\{0,1,2,...,n\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we give the definition of the bridge vertex of a $G$-parking function and obtain an expression of the Tutte polynomial $T_G(x,y)$ of $G$ in terms of $G$-parking functions. We find the Tutte polynomial enumerates the $G$-parking function by the number of the bridge vertices.