On commutative algebra associated to $t$-labeled subforests of a graph

TL;DR

This paper introduces a commutative algebra linked to a graph's $t$-labeled forests, revealing combinatorial properties and connecting algebraic invariants to the Tutte polynomial.

Contribution

It constructs a new algebraic framework for $t$-labeled forests and relates its graded components to the Tutte polynomial, providing novel combinatorial insights.

Findings

Algebra dimension equals the number of $t$-labeled forests

Hilbert polynomial expressed via the Tutte polynomial

Graded component dimensions have combinatorial significance

Abstract

For a given graph $G$, we construct an associated commutative algebra, whose dimension is equal to the number of $t$-labeled forests of $G$. We show that the dimension of the $k$-th graded component of this algebra also has a combinatorial meaning and that its Hilbert polynomial can be expressed through the Tutte polynomial of $G$.