"K-theoretic" analog of Postnikov-Shapiro algebra distinguishes graphs

TL;DR

This paper introduces a K-theoretic analog of a graph-associated algebra, demonstrating it uniquely identifies graphs up to isomorphism and exploring its generalizations.

Contribution

It defines a new K-theoretic algebra for graphs and proves it distinguishes non-isomorphic graphs, extending the algebraic framework of Postnikov-Shapiro.

Findings

The K-theoretic algebra is a complete graph invariant.

Isomorphism of these algebras implies graph isomorphism.

The algebra's generalizations encompass the original algebra.

Abstract

In this paper we study a filtered "K-theoretical" analog of a graded algebra associated to any loopless graph G which was introduced in \cite{PS}. We show that two such filtered algebras are isomorphic if and only if their graphs are isomorphic. We also study a large family of filtered generalizations of the latter graded algebra which includes the above "K-theoretical" analog.