The Hopf Algebra of graph invariants

TL;DR

This paper introduces a Hopf algebra framework for analyzing simple graph isomorphism, using algebraic relations and subgraph counting to develop a new criterion for graph isomorphism testing.

Contribution

It presents a novel algebraic approach using Hopf algebras to address the graph isomorphism problem, connecting algebraic relations with subgraph occurrence counting.

Findings

Algebraic relations can determine graph isomorphism.

Subgraph occurrence counting is effective for isomorphism testing.

The Hopf algebra structure provides a new perspective on graph invariants.

Abstract

We propose an algebraic study of the simple graph isomorphism problem. We define a Hopf algebra from an explicit realization of its elements as formal power series. We show that these series can be evaluated on graphs and count occurrences of subgraphs. We establish a criterion for the isomorphism test of two simple graphs by means of occurrence counting of subgraphs. This criterion is deduced from algebraic relations between elements of our algebra.