# A Hopf algebra on subgraphs of a graph

## TL;DR

This paper constructs a Hopf algebra structure on subgraphs of a graph, explores its dual, and studies algebra morphisms induced by graph homomorphisms, establishing a functor between graph and algebra categories.

## Contribution

It introduces a novel Hopf algebra framework on subgraphs and analyzes its dual and functorial properties, linking graph theory and algebra.

## Key findings

- Established a Hopf algebra on subgraphs
- Derived the dual Hopf algebra structure
- Created a covariant functor from graph to algebra categories

## Abstract

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category.