On the polymatroid Tutte polynomial

TL;DR

This paper explores the polymatroid Tutte polynomial, extending classical matroid invariants, and investigates its properties and coefficients, generalizing hypergraph polynomial results.

Contribution

It proves interpolation properties of the polymatroid Tutte polynomial and analyzes high-order term coefficients, extending known hypergraph polynomial results.

Findings

Proves interpolation of $ ext{J}_P(x,t)$ and $ ext{J}_P(t,y)$ for fixed $t \\geq 1$.

Studies coefficients of high-order terms in $ ext{J}_P(x,1)$ and $ ext{J}_P(1,y)$.

Generalizes results on interior and exterior polynomials of hypergraphs.

Abstract

The Tutte polynomial is a well-studied invariant of matroids. The polymatroid Tutte polynomial $\mathcal{J}_{P}(x,y)$, introduced by Bernardi et al., is an extension of the classical Tutte polynomial from matroids to polymatroids $P$. In this paper, we first prove that $\mathcal{J}_{P}(x,t)$ and $\mathcal{J}_{P}(t,y)$ are interpolating for any fixed real number $t\geq 1$, and then we study the coefficients of high-order terms in $\mathcal{J}_{P}(x,1)$ and $\mathcal{J}_{P}(1,y)$. These results generalize results on interior and exterior polynomials of hypergraphs.