Csikv\'{a}ri's poset and Tutte polynomial

TL;DR

This paper confirms that Csikvári's poset on trees can be used to identify extremal values of the Tutte polynomial for cones over trees, strengthening previous results and conjectures.

Contribution

It proves the conjecture that Csikvári's poset characterizes extremal Tutte polynomial values for cones over trees, extending prior work.

Findings

Csikvári's poset accurately predicts extremal Tutte polynomial values.

The conjecture by Reiner and Smith is proven true.

The result applies to cones over trees and their Tutte polynomials.

Abstract

Csikv\'{a}ri constructed a poset on trees to prove that several graph functions attain extreme values at the star and the path among the trees on a fixed number of vertices. Reiner and Smith proved that the Tutte polynomials $T(1,y)$ of cones over trees, which are the graphs obtained by attaching a cone vertex to a tree, have the described extreme behavior. They further conjectured that the result can be strengthened in terms of Csikv\'{a}ri's poset. We solve this conjecture affirmatively.