On an ordering-dependent generalization of Tutte polynomial

TL;DR

This paper introduces an order-dependent generalization of the Tutte polynomial, motivated by stochastic differential equations, which reduces to the known Tutte-Fortuin-Kasteleyn polynomial under certain limits.

Contribution

It proposes a new combinatorial invariant that depends on the order of contraction-deletion operations, extending the classical Tutte polynomial for applications in stochastic calculus.

Findings

The order-dependent Tutte polynomial is defined and analyzed.

In the limit of control parameters, it converges to the multivariate Tutte-Fortuin-Kasteleyn polynomial.

Provides a new perspective linking graph invariants with stochastic processes.

Abstract

A generalization of Tutte polynomial involved in the evaluation of the moments of the integrated geometric Brownian in the Ito formalism is discussed. The new combinatorial invariant depends on the order in which the sequence of contraction-deletions have been performed on the graph. Thus, this work provides a motivation for studying an order-dependent Tutte polynomial in the context of stochastic differential equations. We show that in the limit of the control parameters encoding the ordering going to zero, the multivariate Tutte-Fortuin-Kasteleyn polynomial is recovered.