Coefficients of the Tutte polynomial and minimal edge cuts of a graph

Abstract

Let G be a $(k+1)$-edge connected graph with order n and size m. From a general result on the coefficients of polymatroid Tutte polynomial, Guan et al. (2023) [16] derived that$\left[y^{g-i}\right]T_G(1,y)=\left(\begin{array}{c}n+i-2\\ n-2\end{array}\right),0\le i\le k,$ where $T_G(x,y)$ is the Tutte polynomial of G and $g=m-n+1$. Recall that the coefficients of $T_G(1,y)$ have many combinatorial explanations, including spanning trees, parking functions, superstable configurations (or recurrent configurations) of the Abelian Sandpile Model (ASM), and so on. Here we find that the above result has a simple and direct proof in terms of the superstable configurations of ASM. Motivated by this, in this paper, by constructing mappings between different sets, we first establish a relationship between non-superstable configurations and minimal edge cuts of G, then we generalize the above result from $0\le i\le k$ to $0\le i<\frac{3(k+1)}{2}$. In precise,$\left[y^{g-i}\right]T_G(1,y)=\left(\begin{array}{c}n+i-2\\ n-2\end{array}\right)-\sum _{j=k+1}^i\left(\begin{array}{c}n+i-j-2\\ n-2\end{array}\right)\left|{\mathrm{EC}}_j\right(G\left)\right|,0\le i<\frac{3(k+1)}{2},$ where ${\mathrm{EC}}_j\left(G\right)$ denotes the set of all minimal edge cuts with j edges. Thus, our results provide a new combinatorial explanation for some coefficients of $T_G(1,y)$ in terms of minimal edge cuts.