Some inequalities for the Tutte polynomial

TL;DR

This paper investigates inequalities and convexity properties of the Tutte polynomial for specific classes of matroids and graphs, providing new bounds and conjectures related to the polynomial's behavior.

Contribution

It proves convexity of the Tutte polynomial along certain lines for coloopless paving matroids and establishes key inequalities, including a conjecture for these matroids.

Findings

Convexity of the Tutte polynomial along line segments in the positive quadrant.

Inequality T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2 in certain matroids.

Proof of a conjecture for some graph and matroid families.

Abstract

We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two disjoint bases or whose ground set is the union of two bases of M*. For this latter class we give a proof that T_M(a,a) <= max {T_M(2a,0), T_M(0,2a)} for a >= 2. We conjecture that T_M(1,1) <= max {T_M(2,0), T_M(0,2)} for the same class of matroids. We also prove this conjecture for some families of graphs and matroids.