Signatures of graphs for bicommutative Hopf algebras

TL;DR

This paper explores how signature functionals over combinatorial Hopf algebras of graphs can unify various subgraph counting methods through algebraic identities and isomorphisms.

Contribution

It demonstrates that different graph substructure Hopf algebras are isomorphic to a polynomial Hopf algebra, linking algebraic identities to counting subgraphs.

Findings

Algebraic identities correspond to character properties and Chen's identity.

Different notions of subgraphs lead to isomorphic polynomial Hopf algebras.

Isomorphisms respect counting operations across algebraic frameworks.

Abstract

This article approaches the counting of subgraphs, in terms of signature-type functionals defined over combinatorial Hopf algebras of graphs. Well-known algebraic identities that arise in the context of counting subgraphs are then captured by their character property and a type of "Chen's identity". While different notions of subgraphs (and homomorphisms) correspond to different combinatorial Hopf algebras on graphs, we will show that they are all isomorphic to a polynomial Hopf algebra. In addition, the isomorphy between the Hopf algebras can be realized by maps that respect the counting operations.