The incidence Hopf algebra of graphs

TL;DR

This paper introduces a new, simplified formula for the antipode in the graph algebra Hopf algebra, leveraging acyclic orientations, and explores applications to Tutte polynomial evaluations.

Contribution

It provides a more efficient antipode formula for the graph algebra and demonstrates its applications to Tutte polynomial evaluations.

Findings

New antipode formula with fewer terms

Simplified combinatorial description of the Hopf algebra

Applications to Tutte polynomial evaluations

Abstract

The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.