# Several extreme coefficients of the Tutte polynomial of graphs

## TL;DR

This paper derives explicit formulas for eight extreme coefficients of the Tutte polynomial of certain graphs, linking them to subgraph structures and related polynomials, and explores their duality and Jones polynomial specializations.

## Contribution

It provides new explicit expressions for specific extreme coefficients of the Tutte polynomial in terms of subgraph structures, expanding understanding of their combinatorial significance.

## Key findings

- Explicit formulas for eight extreme coefficients of the Tutte polynomial.
- Connections between Tutte coefficients and chromatic/flow polynomial coefficients.
- Discussion of duality and Jones polynomial specializations.

## Abstract

Let $t_{i,j}$ be the coefficient of $x^iy^j$ in the Tutte polynomial $T(G;x,y)$ of a connected bridgeless and loopless graph $G$ with order $n$ and size $m$. It is trivial that $t_{0,m-n+1}=1$ and $t_{n-1,0}=1$. In this paper, we obtain expressions of another eight extreme coefficients $t_{i,j}$'s with $(i,j)=(0,m-n)$,$(0,m-n-1)$,$(n-2,0)$,$(n-3,0)$,$(1,m-n)$,$(1,m-n-1)$,$(n-2,1)$ and $(n-3,1)$ in terms of small substructures of $G$. Among them, the former four can be obtained by using coefficients of the highest, second highest and third highest terms of chromatic or flow polynomials, and vice versa. We also discuss their duality property and their specializations to extreme coefficients of the Jones polynomial.